US20200276493A1 - Districting Strategy Game - Google Patents

Abstract

Games of the present technology are strategic districting games in which one or more players are presented with a region divided into sectors, where each sector has a given set of elements, and players are tasked with combining the sectors to create a plurality of districts that achieve an objective.

Description

RELATED APPLICATIONS
• This application claims priority to U.S. Provisional Application Ser. No. 62/785,824, filed on Dec. 28, 2018, the entirety of which is incorporated by reference herein.
FIELD OF THE INVENTION
• The present technology relates to the field of games, and more particularly to strategy games.
COPYRIGHT NOTICE
• A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyright rights whatsoever.
SUMMARY
• Strategy games of the present technology may be played by at least one player. The playing surface comprises a region that includes a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector. Each sector contains a set of elements, each element of the set of elements having a type and quantity. During play, each player makes one move per turn, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.
• Methods of playing strategy games of the present technology include providing a playing surface that includes a region comprising a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector, and each sector contains a set of elements, each element of the set of elements having a type and quantity. Methods of playing strategy games of the present technology further include making one move per turn per player, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.
BRIEF DESCRIPTION OF THE DRAWINGS
• Specific examples have been chosen for purposes of illustration and description, and are shown in the accompanying drawings, forming a part of the specification. Within the Figures, like parts have been given like numbers for ease of reference. It should be understood that the drawings are not necessarily drawn to scale and that they are intended to be merely illustrative.
• FIG. 1 illustrates a first example of a strategy game of the present technology, having a first region.
• FIG. 2 illustrates an example of a region that can be included in a second strategy game of the present technology.
• FIG. 3 illustrates examples of possible district shapes having four sectors that can be formed during play of a strategy game using the region of FIG. 2.
• FIG. 4 illustrates a possible solution of a strategy game using the region of FIG. 2 in accordance with a first pre-defined goal.
• FIG. 5 illustrates a possible solution of a strategy game using the region of FIG. 2 in accordance with a second pre-defined goal.
• FIG. 6 illustrates a possible solution of a strategy game using the region of FIG. 2 in accordance with a third pre-defined goal.
• FIG. 7 illustrates examples of possible district shapes having six sectors that can be formed during play of a strategy game using the region of FIG. 1.
• FIG. 8 illustrates a possible solution of a strategy game using the region of FIG. 1 in accordance with a first pre-defined goal.
• FIG. 9 illustrates a possible solution of a strategy game using the region of FIG. 1 in accordance with a second pre-defined goal.
• FIG. 10 illustrates a possible solution of a strategy game using the region of FIG. 1 in accordance with a third pre-defined goal.
• FIG. 11 illustrates an example of a region that can be included in a third strategy game of the present technology.
• FIG. 12 illustrates a strategy game including the region of FIG. 11.
• FIG. 13 illustrates the scoreboard of FIG. 12.
• FIG. 14 illustrates one example of sector tiles that can be used in a strategy game of FIG. 12.
• FIG. 15A illustrates a possible arrangement of markers on a region during play.
• FIG. 15B illustrates a second possible arrangement of markers on a region during play.
• FIG. 16A illustrates a third possible arrangement of markers on a region during play.
• FIG. 16B illustrates a fourth possible arrangement of markers on a region during play.
• FIG. 17A illustrates a fifth possible arrangement of markers on a region during play.
• FIG. 17B illustrates a sixth possible arrangement of markers on a region during play.
• FIG. 18 illustrates examples of sector relationships.
• FIG. 19 illustrates a seventh possible arrangement of markers on a region during play.
• FIG. 20 illustrates an eighth possible arrangement of markers on a region during play.
• FIG. 21 illustrates one possible position for the game of FIG. 11 after 15 turns.
• FIG. 22 illustrates one possible position for the game of FIG. 21 after 23 turns.
• FIG. 23 illustrates one possible position for the game of FIG. 22 after 27 turns.
• FIG. 24 illustrates one possible position for the game of FIG. 23 after 29 turns.
• FIG. 25 illustrates one possible position for the game of FIG. 24 after 32 turns.
• FIG. 26 illustrates one possible position for the game of FIG. 25 after 34 turns.
• FIG. 27 illustrates one possible position for the game of FIG. 26 after 35 turns.
• FIG. 28 illustrates one possible position for the game of FIG. 27 after 36 turns.
• FIG. 29 illustrates one possible position for the game of FIG. 28 after 37 turns.
• FIG. 30 illustrates one possible position for a final position of the game of FIG. 29.
• FIG. 31 illustrates a guide for arrangement of a symmetric game of FIG. 11.
• FIG. 32 illustrates a symmetric game setup table for a symmetric game of FIG. 11.
• FIG. 33 illustrates a fourth example of a strategy game of the present technology.
• FIG. 34 illustrates one possible initial position for a strategy game of FIG. 33.
• FIG. 35 illustrates one possible final position for a strategy game of FIG. 34.
• FIG. 36 illustrates a fifth example of a strategy game of the present technology.
• FIG. 37 illustrates a scoreboard that may be used with the strategy game of FIG. 36.
• FIG. 38A illustrates a possible solution by a first player in phase one of a first strategy game of FIG. 36.
• FIG. 38B illustrates a possible solution by a second player in phase one of the strategy game of FIG. 38A.
• FIG. 39A illustrates a possible solution by a first player in phase two of the strategy game of FIG. 38A.
• FIG. 39B illustrates a possible solution by a second player in phase two of the strategy game of FIG. 38A.
• FIG. 40A illustrates a possible solution by a first player in a second strategy game of FIG. 36.
• FIG. 40B illustrates a possible solution by a second player in the strategy game of FIG. 40A.
• FIG. 41 illustrates examples of sector tiles in a strategy game of the present technology having four elements.
• FIG. 42A illustrates an initial position in a region of a sixth example of a strategy game of the present technology.
• FIG. 42B illustrates one possible final position for a strategy game of FIG. 42A.
• FIG. 43 illustrates a scoreboard that may be used in the strategy game of FIG. 42A.
• FIG. 44 illustrates a seventh example of a strategy game of the present technology.
• FIG. 45 illustrates one possible initial position for a strategy game of FIG. 44.
• FIG. 46 illustrates one possible final position for a strategy game of FIG. 45.
• FIG. 47 illustrates one possible final position for an eighth example of a strategy game of the present technology.
• FIG. 48 illustrates a scoreboard that may be used in a strategy game of FIG. 47.
• FIG. 49 illustrates a game board that may be used with a ninth example of a strategy game of the present technology.
• FIG. 50 illustrates one possible initial position for a tenth example of a strategy game of the present technology.
• FIG. 51 illustrates examples of sector tiles that may be used in a strategy game of FIG. 50.
• FIG. 52 illustrates a scoreboard that may be used in a strategy game of FIG. 50.
• FIG. 53 illustrates a rotationally symmetric sector arrangement for a strategy game of FIG. 50.
• FIG. 54 illustrates one possible region that may be used in an eleventh example of a strategy game of the present technology.
• FIG. 55 illustrates examples of sector tiles that may be used in a strategy game of FIG. 54.
• FIG. 56 illustrates one possible initial position in a region that may be used in a twelfth example of a strategy game of the present technology.
• FIG. 57 illustrates part 1 of scoreboard that may be used in the strategy game of FIG. 56.
• FIG. 58 illustrates part 2 of scoreboard that may be used in the strategy game of FIG. 56.
• FIG. 59 illustrates one possible position for the game of FIG. 56 during play.
• FIG. 60 illustrates one possible position for the game of FIG. 59 after 40 turns.
• FIG. 61 illustrates one possible position for the game of FIG. 60 after 41 turns.
• FIG. 62 illustrates one possible position for the game of FIG. 61 after 42 turns.
• FIG. 63 illustrates one possible position for the game of FIG. 62 after 65 turns.
• FIG. 64 illustrates one possible position for the game of FIG. 63 after 70 turns.
• FIG. 65 illustrates one possible position for the game of FIG. 64 after 75 turns.
• FIG. 66 illustrates one possible position for the game of FIG. 65 after 79 turns.
• FIG. 67 illustrates one possible position for the game of FIG. 66 after 81 turns.
• FIG. 68 illustrates one possible position for the game of FIG. 67 after 83 turns.
• FIG. 69 illustrates one possible position for the game of FIG. 68 after 84 turns.
• FIG. 70 illustrates one possible final position for phase 1 the game of FIG. 69.
• FIG. 71 illustrates one possible initial position in a region that may be used in a thirteenth example of a strategy game of the present technology.
• FIG. 72 illustrates part 1 of scoreboard that may be used in the strategy game of FIG. 71.
• FIG. 73 illustrates part 2 of scoreboard that may be used in the strategy game of FIG. 71.
• FIG. 74 illustrates one possible final position for the strategy game of FIG. 71.
• FIG. 75 provides a table that includes a key to symbols used in FIGS. 1-74.
DETAILED DESCRIPTION
• The present technology provides strategy games involving the division of a region into districts to achieve one or more predefined goals. The use of the prefix “pre” herein means any time prior to beginning the play of the game. Games of the present invention can be provided as board games, sector based games, paper based games, electronic games, or games on any other suitable presentation medium. For example, the playing surface on which a region may be provided may take the form of a board, a set of sectors, a piece of paper, a three-dimensional form, or a screen.
• Table 1, provided in FIG. 75, is a key that explains the meaning of the various numbers and shapes that appear in the Figures. The first 5 rows of Table 1 represent designations that may be used with sectors. The first row of Table 1 is a sector number, and each sector within a region may have a distinct sector number to distinguish the sector from any other sector within the region. The fifth row represents the total population within a sector. The second through fourth rows represent designations of a party majority, for use in examples of strategy games that include political parties, which are represented in the Table as including a red party, a blue party, and a green party. The final row of Table 1 provides a designation for a scoring token, which may be used in examples of strategy games that have a scoreboard. The remainder of the rows of Table 1 provide representations of designations for markers, including home base markers and expansion markers, that may be placed onto a sector during a player's move to assign that sector to a district.
• FIG. 1 illustrates one example of a strategy game 100 of the present technology. The strategy game 100 includes a region 102, which is a hexagon. Generally, in strategy games of the present technology, the region has at least one area or volume bounded by pre-defined external boundaries. The region can be any real or imagined bounded two-dimensional shape or three-dimensional form, such as a polygon or other geometric shape, a polyhedron, or a geographical area. In many examples, the region is presented as a map. While the region is preferably defined by a single bounded shape or form, a region can include multiple bounded shapes or forms. In examples where a region has multiple bounded shapes or forms, it is preferred that at least a portion of each bounded shape or form be predefined to be connected with at least part of one of the other bounded shapes or forms.
• The region 102 is pre-divided into a plurality of sectors 104. The sectors are stationary, and do not move during play of the game. Each sector has an area or volume within the region bounded by pre-defined boundaries. Each sector is distinct and does not overlap with any other sector. Each sector can be any real or imagined bounded shape or form—such as a polygon or other geometric shape, a polyhedron or other three-dimensional form, or a geographical area—within the region. For example, each sector 104 is a triangle. For purposes of the strategy games described herein, a sector cannot be further divided. Every part of the region 102 is defined as being part of a sector, and, collectively, the sectors cover the entire area or volume of the region 102.
• Each sector 104 contains a set of elements, each set of elements including one or more elements. The set of elements in a sector is a complete list of the elements contained in the sector, and each element of the set of elements has a type and quantity. The quantity of an element may be any amount, and is preferably greater than zero. In some examples, the type of each element is voters that favor a particular political party. In other examples, the type of each element may be a resource (i.e., something useful), a hazard (i.e., something harmful), or scrap (i.e., something neither useful nor harmful). Each type of element may have the same value to each player. Alternatively, each element may have a value that is player-specific. That is, the same element may be a resource (i.e., something useful) to one player but a hazard (i.e., something harmful) or scrap (i.e., something neither useful nor harmful) to another player.
• Referring to FIG. 1, the illustrated strategy game 100 is a game in which the type of element in each sector is voters that favor a particular political party, and the quantity is the margin of voters within the sector that favor the indicated political party. For example, voters in sector 106 favor the Blue party by a margin of 2, and voters in sector 108 favor the Red party by a margin of 9. The voter margin may represent single voters, hundreds of voters, thousands of voters, or any other suitable amount of voters, as appropriate for a given game.
• The strategy game 100 shown in FIG. 1 also includes a set of markers 110, wherein the set of markers comprises a plurality of marker subsets 112, each marker subset representing a district. Specifically, each marker within each marker subset 112 is configured to be placed on a sector to assign the sector as being part of the district represented by the marker subset 112. Each marker subset 112 comprises a plurality of markers, and may include a home base marker 114 and at least one, preferably more than one, expansion marker 116. In strategy game 100, there are fifty-four sectors 104, and nine marker subsets 112, each marker subset 112 representing one of nine districts will be created during the game. In order for each sector 104 to have a marker on it at the end of the game, there are preferably at least six markers in each marker subset 112. In this example, there may be one home base marker 114 and at least five expansion markers 116 in each marker subset 112.
• The strategy game 100 shown in FIG. 1 also includes a scoreboard 118. The scoreboard 118 has one row for each of the nine districts that will be created during the game, each row containing a scoring spectrum, such as the illustrated spectrum of which party (Red or Blue) is favored and the margin by which the party is favored within the district. Each row has a scoring token 120 that may be used keep track of the favored party and total voter margin within each district.
• Games of the present technology may be played by at least one player. Some embodiments are designed to be played by a single player, while other embodiments are designed to be played by a plurality of players, such as at least two players. The term “player” as used herein can mean one individual, or a team of individuals. In many games of the present technology, two or more players take alternating turns, and each player must make one move per turn. In some embodiments of the present technology there is only one player who faces an individual challenge. In some games of the present technology, two or more players independently consider the exact same challenge, each using a separate copy of identical game components. In such examples, each player may take turns independently of the other players, using their own set of the game components. Turns may or may not be under time constraints. The player who does the best job of attaining the pre-defined goal wins the game.
• Generally, in order to play a game of the present technology, one or more players are provided with a region that has been divided into non-overlapping sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the area or volume of the large region. At the outset, the one or more players are informed of each sector's precise shape, location, and set of elements, and such information may be depicted graphically. In most examples, there are no districts yet created within the region at the start of play, and each sector is initially considered to be unassigned.
• During play, each player makes one move per turn by assigning a sector to a district, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district. Examples of pre-defined goals include, but are not limited to: (a) maximizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; (b) minimizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; and (c) maximizing a number of points earned by at least one of the players, where the number of points earned by the at least one player depends on the aggregation of the elements within each district. In examples of strategy games of the present technology that include at least two political parties, such as strategy game 100 of FIG. 1, examples of pre-defined goals include, but are not limited to: (a) maximizing a portion of the given number of districts controlled by one of the political parties, and (b) equalizing a portion of the given number of districts controlled by each of the political parties.
• Rules of strategy games of the present technology define types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors. For example, the rules may provide at least two general categories of moves that a player can make:
• • 1. Establish a new district by assigning the first sector that belongs to it.
  • 2. Expand a district by assigning an unassigned sector to an already established district.
• In some examples, the rules include additional categories of moves that are permitted. For example, three additional categories of moves could be:
• • 3. Reassign a sector from one district to another district.
  • 4. Break up one or more adjacent districts by un-assigning all sectors that belong to them.
  • 5. Freeze a district. No player may modify the district during the next several turns.
• As another example, rules regarding how districts may be formed may include:
• • A. Every sector within the region must be assigned to exactly one district.
  • B. Each district must be a single connected piece.
  • C. Each district may be required to have at least, or at most, a certain number of sectors, or may be required to include a specific pre-defined number of sectors.
• In many games of the present technology, play concludes when there are no more permitted moves, or when every sector within the region has been assigned to a district. If there are two or more players, the winning player is the player that does the best job of achieving the pre-defined goal. If there is one player, the player wins if the pre-defined goal is achieved, and otherwise loses.
• Computer games incorporating the concept may be played in at least two modes. In mode 1, a computer plays the role of (i.e., makes the decisions for) one or more players. In this mode, the computer makes use of sophisticated artificial intelligence techniques that are programmed into it ahead of time by a team of expert computer scientists. In mode 2, the computer provides visualization, data storage, and communication services to facilitate play but does not participate as a decision maker during play.
Taxonomy of Strategy Games of the Present Technology:
• Strategy games of present technology include hundreds of recreational and non-recreational games and puzzles. A taxonomy for the exemplary strategy games is provided below, and is based on a nine-part code. This code summarizes the main aspects of a given example of the present technology. Each part of the code contains one or more capital letters or integers and is separated from the other parts of the code by forward slashes. The generic code for an example of the current technology is as follows:
• Part1/Part2/Part3/Part4/Part5/Part6/Part7/Part8/Part9
• Part 1 of the code is either the letter “A” or “D.” It is “A” if the example is analog in nature; it is “D” if the example is digital in nature.
• Part 2 is either “Z” or “G.” It is “Z” if the example is a single-player puzzle (e.g., an individual challenge like a Sudoku puzzle). It is “G” if the example is a multi-player game.
• Part 3 refers to the shape of each sector and the number of sectors. It consists of a letter followed by an integer with no interceding punctuation. It begins with “S” if each sector is a square; “T” if each sector is an equilateral triangle; “H” if each sector is a regular hexagon; “C” if each sector is a complex, two-dimensional shape such as the shape of a real-world county or country; and “0” if each sector is another shape (e.g., a three-dimensional form). The integer Y that follows the letter indicates how many sectors are in the game. If the letter is (S, T, H), the number of sectors is (36*Y, 54*Y, 37*Y) respectively or slightly less than this. If the letter is “C” or “0,” the value Y gives the exact number of sectors.
• Part 4 is either “P” or “N.” It is “P” if the example focuses on politics. It is “N” if the example does not focus on politics.
• Part 5 is an integer that gives the number of element types that are found within the sectors. Its value often ranges from 2-6.
• Part 6 is either the letter “V” or an integer. If it is the letter “V,” the number of districts to be formed is variable and is unknown at the start. Otherwise, the number of districts to be formed is known at the start and equals the value in this part of the code.
• Part 7 specifies the game paradigm. It only applies to multi-player games, thus it exists only if part 2 of the code is “G.” Part 7 is “U” if the game involves alternating, turn-based play. It is “I” if the game involves simultaneous independent play in which each player takes turns independently of the others. Games with simultaneous independent play can have any number of players, whereas games with alternating, turn-based play typically have no more than 6 players.
• Parts 8 and 9 only apply to multi-player games with alternating, turn-based play; these parts of the code exist only if part 2 of the code is “G” and part 7 of the code is “U.”
• Part 8 indicates the number of players in the game. It is expressed as a range—with two integers separated by a hyphen—if different numbers of players can play the game. It is a single integer if the game is designed for a specific number of players (e.g. for two players only).
• Part 9 contains one or more of the letters “E,” “X,” “R,” “B,” and “F.” These five letters respectively refer to five categories of allowed moves—“Establish,” “Expand,” “Reassign,” “Break up,” and “Freeze”—which are briefly described in a previous paragraph. This part of the code contains the letters that correspond to the categories of moves that are allowed in the game.
• The description of each example provided below begins with a discussion of its taxonomic code. This code gives the reader a quick understanding of the example's main aspects. One or more parts of a code may contain the question mark symbol “?” if those aspects are unspecified.
EXAMPLES
• Several non-limiting examples of strategy games of the present technology are provided below. While the examples use numbers, letters, and generic shapes to distinguish between different districts, element types, and sectors, it should be noted that other methods of distinction could be used. For example, colors or specialized graphics could be used.
Example 1
• Strategy games of the Example 1 have a taxonomic code A/Z/S1/P/2/9. They are analog, single-player puzzles with square sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed. A nearly unlimited number of possible instances of this kind of puzzle can be created, one of which is illustrated in FIG. 2 as strategy game 200. A collection of instances of this kind of strategy game can be assembled in a booklet.
• Two aspects distinguish this kind of puzzle from most other types of logic puzzles. First, there may be multiple solutions to a given puzzle; a unique solution is not guaranteed. Second, the “partial solution” concept does not apply. In other words, if one partially finishes a puzzle, there is no guarantee that the partial solution will give rise to a complete solution. A logical “guess and check” approach is recommended for solving this kind of puzzle.
• In strategy game 200, the player is given a map of a square shaped region 202 that has been divided into 36 square shaped sectors 204, which are arranged in six rows and six columns. The player is tasked with dividing the region 200 into a given number of political districts—i.e. to draw lines that define the boundaries of the districts—in order to achieve the stated objective.
• Two types of elements—two political parties—occupy the region. One element is the Red Party and the other element is the Blue Party. Each sector 204 may represent a community, and each community has a number which is the community's voter margin. A black number in a white circle means that there are more Red Party supporters than Blue Party supporters in the community (see the key in Table 1). In such a case, the community favors the Red Party. A white number in a black circle means that there are more Blue Party supporters than Red Party supporters in the community. In such a case, the community favors the Blue Party. The number itself is the margin (in thousands of voters) by which the community supports one party over the other. For example, sector 206 has a black 3 in a white circle, which may mean that there are 3000 more Red Party supporters than Blue Party supporters in that community. In this case, the community's voter margin is “+3 Red.” In sector 208, there is a white 4 in a black circle, which may mean that there are 4000 more Blue Party supporters than Red Party supporters in that community. The voter margin in sector 208 is “+4 Blue.” In Sector 210, there is a zero, which means that the community equally supports the two parties. In such a case, the community's voter margin is 0.
• In each puzzle of Example 1, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts.
• The rules for forming political districts are as follows.
• • 1. Each sector must belong in its entirety to one and only one district.
  • 2. No two districts may overlap.
  • 3. Each district must consist of four sectors that form a single connected piece.
• FIG. 3 shows examples of shapes 302, 304, 306, 308, and 310, in which a district may be formed. Each shape of FIG. 3 is formed from four sectors 204 that form a single connected piece. Rotations and reflections of shapes 302-310 would also be acceptable.
• When a district is formed, the player must pay attention to its voter margin (i.e. margin). A district's voter margin indicates which party has more voters in the district. A district's voter margin depends on the voter margins of the sectors in the district. It equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district.
• For example, if a district has four sectors with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district's voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district's voter margin is 0.
• In each puzzle, the player may be asked to pursue one of three goals:
• • A. Create political districts that maximize the advantage of the Red Party
  • B. Create political districts that maximize the advantage of the Blue Party
  • C. Create political districts that equalize the advantage of the two parties
• Goals A, B, and C relate to the voter margins of the districts that are formed. In exact terms, Goal A is to, first and foremost, maximize the number of districts controlled by the Red Party and, secondarily, maximize the margin by which the Red Party controls its least safe district. Goal B is to do the same except to the benefit of the Blue Party. Goal C is to (i) equalize the number of districts controlled by each party, (ii) equalize the margin by which each party controls its least safe district, and (iii) maximize the number of tied districts that have a voter margin of 0. A party's least safe district is the district in which it has the smallest majority.
• Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent.
• There are different solutions to the strategy game 200 for each goal A, B, and C. For example, FIG. 4 shows a solution in which the region 202 has been divided into nine districts 212-228 to achieve goal A. In the solution of FIG. 4, the Red party controls seven of the districts by a margin of at least +2. These include districts 212, 214, 218, 220, 222, 224, and 228. In FIG. 5, the region 202 has been divided into nine districts 230-246 to achieve goal B. In the solution of FIG. 5, the Blue party controls seven of the districts by a margin of at least +2. These include districts 230, 232, 234, 240, 242, 244, and 246. In FIG. 6, the region 200 has been divided into nine districts 248-264 to achieve goal C. In the solution of FIG. 6, each party controls two districts, five districts are tied, and the margin by which each party controls its least safe district is +9. Districts 248, 250, 252, 254, and 262 are tied. Districts 256 and 260 are controlled by the Red Party and have voter margins of “+9 Red” and “+12 Red” respectively. Districts 258 and 264 are controlled by the Blue Party and have voter margins of “+12 Blue” and “+9 Blue” respectively.
Example 2
• Strategy games of the Example 2 have a taxonomic code A/Z/T1/P/2/9. They are analog, single-player puzzles with triangular sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed.
• In some examples of this kind of puzzle, the region may be a hexagonal region that has 54 triangular sectors (i.e. communities). Each community may have the same population. An example of such a region is region 102 in FIG. 1.
• In the example shown in FIG. 1, supporters of two political parties—Red and Blue—occupy the region 102. Each sector 104 represents a community, and has a number which represents the community's voter margin. A black number in a white circle means that there are more Red Party supporters than Blue Party supporters in the community. A white number in a black circle means that there are more Blue Party supporters than Red Party supporters in the community. The number within each sector is the margin (in thousands of voters) by which the community supports one party over the other.
• In each puzzle of Example 2, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts.
• The rules for forming political districts are as follows.
• • 1. Each community must belong in its entirety to one and only one district.
  • 2. No two districts may overlap.
  • 3. Each district must consist of six sectors that form a single connected piece.
• FIG. 7 shows examples of shapes 402-424 in which a district may be formed. Each shape of FIG. 7 is formed from six sectors 104 that form a single connected piece. Rotations and reflections of shapes 402-424 would also be acceptable.
• When a district is formed, the player must pay attention to its voter margin. Just as in Example 1, a district's voter margin equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district.
• In each puzzle, the player may be asked to pursue one of three goals:
• • A. Create political districts that maximize the advantage of the Red Party
  • B. Create political districts that maximize the advantage of the Blue Party
  • C. Create political districts that equalize the advantage of the two parties
• Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent.
• There are different solutions to the strategy game 100 for each goal A, B, and C. For example, FIG. 8 shows a solution in which the region 102 has been divided into nine districts 122-138 to achieve goal A. In the solution of FIG. 8, the Red party controls eight of the districts—all except district 134—by a margin of at least +1. In FIG. 9, the region 102 has been divided into nine districts 140-156 to achieve goal B. In the solution of FIG. 9, the Blue party controls eight of the districts—all except district 146—by a margin of at least +1. In FIG. 10, the region 102 has been divided into nine districts 158-174 to achieve goal C. In the solution of FIG. 10, each party controls three districts, three districts are tied, and the margin by which each party controls its least safe district is +2. Districts 162, 172, and 174 are tied. Districts 160, 164, and 166 are controlled by the Red Party and have voter margins of “+2 Red,” “+6 Red,” and “+17 Red” respectively. Districts 158, 168, and 170 are controlled by the Blue Party and have voter margins of “+12 Blue,” “+2 Blue,” and “+11 Blue” respectively.
Example 3
• Strategy games of the Example 3 have a taxonomic code A/G/S1/P/2/9/U/2/EXR. They are analog, multi-player games with square sectors and a political focus in which two types of elements—namely two political parties—are present and nine districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• FIGS. 11-32 illustrate a strategy game 500, having a region 502. In strategy game 500, two players representing opposing political parties—Red and Blue—vie for political control of a square state (region 502) by competitively creating nine political districts out of 36 square sectors 504 in alternating, turn-based fashion. FIG. 11 shows the initial arrangement of the region 502, prior to being divided into districts by the game play. FIG. 30 shows one possible final arrangement of the region 502, after the region 502 is divided into nine districts by game play.
• In strategy game 500, each sector represents a community. The region represents a state and has an American-style, two-party political system in which one person is elected to represent each political district. At the start, the districts have not been formed and the players know the location and political composition of each sector (i.e., which party its citizens favor and by how much). During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time, in alternating turns. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated, which may be done by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game.
• In the final position, shown in FIG. 30, home base markers 600 and expansion markers 602, of the types shown in Table 1, are shown that have been placed by the players during the game. Bold lines indicate boundaries between the nine districts 510-526.
• The table below shows the final result of this game. The Blue Party wins this example game by a score of 5 districts to 3 districts (with one tied district):

• 		Party in	By How
  	District	Control	Much

  	Brown	Neither	0	(=6 + 3 − 5 − 4)
  	Red	Blue	7	(=8 + 6 + 0 − 7)
  	Orange	Blue	3	(=5 + 3 − 3 − 2)
  	Yellow	Red	8	(=9 + 5 − 2 − 4)
  	Green	Red	5	(=7 + 6 − 8)
  	Blue	Blue	2	(=9 + 2 − 1 − 8)
  	Purple	Red	9	(=1 + 2 + 8 − 1 − 1)
  	Pink	Blue	1	(=7 + 4 − 6 − 4)
  	Gray	Blue	9	(=0 + 5 + 7 − 3)

Game Components
• In strategy games of this example, the region 502 is laid out and includes boundaries for the sectors. To allow multiple varied games to be played, however, the elements of the sectors are not pre-printed on the region. Instead, sector tiles 610 (FIG. 14) are provided, which may be shuffled or otherwise reorganized, and laid onto the region at the start of a game. The set of elements 616 for each sector tile 610 is provided on the sector tile. FIGS. 12-14 illustrate game components that may be used in strategy game 500:
• • A region 502, configured to receive sector tiles 610. In this example, the region is a 6×6 square, forming a 36 square grid, each square of the grid being numbered for reference from 1 to 36, and being configured to receive a sector tile 610.
  • 36 square sector tiles 610, examples of two types of which are shown in FIG. 14 as sector tiles 612 and 614. The set of elements 616 on each sector tile 610 are shown as being marked twice on each sector tile 610, in different orientations for visibility from different angles, but it should be understood that the set of elements on a sector tile 610 may be shown in any suitable manner, such as a single representation as shown in FIG. 11. The set of elements that may be used for the 36 sector tiles 610 for strategy game 500 are listed below, and the quantity of each sector type is shown in parentheses:

• +9 Red (1)	+5 Red (2)	+1 Red	(2)	+3 Blue (2)	+7 Blue (2)
  +8 Red (2)	+4 Red (2)	0	(2)	+4 Blue (2)	+8 Blue (2)
  +7 Red (2)	+3 Red (2)	+1 Blue	(2)	+5 Blue (2)	+9 Blue (1)
  +6 Red (2)	+2 Red (2)	+2 Blue	(2)	+6 Blue (2)

• • 9 home base markers 600 (one for each marker subset representing one of nine districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)
  • 162 expansion markers 602 (eighteen for each marker subset representing one of the nine districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray)
  • 9 scoring tokens 606 (one for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each which may be cube-shaped with faces showing the numbers 10, 20, 30, 40, and 50 or any other suitable shape
  • 2 ten-sided dice 618 (1 black and 1 white), each showing the values 0-9
  • Scoreboard 604 (FIG. 13), which may have nine rows 608, one for each of the nine districts. Each row 608 may contain a scoring spectrum that indicates which party (Red or Blue) is favored and the margin by which the party is favored within the district.
Game Setup
• Players may decide (yes or no) if a symmetric game will be played and (yes or no) if phase 2 of the game will be played. The “no-no” option is recommended for beginners. The “yes-no” option is a game of pure skill, whereas the “no-yes” option maximizes the role of luck in the game. Players then decide who plays Red, who plays Blue, and who takes the first turn.
• The game components may be laid out at the start of the game is shown in FIG. 12. The sector tiles 610 may be drawn one at a time and placed face up in a grid square of the region 502 to form a sector 504. The sector tiles 610 may be placed onto the region sequentially on grid squares 1-36, or in any other suitable manner. The resulting region 502 with its sectors 504 may look like the region represented in FIG. 11.
Sector Tiles
• Each sector tile 610 has a set of elements marked at least once thereon that show the element type (Red Party or Blue Party being favored) and the quantity (voter margin). A black number in a white circle, such as first element set 506 in FIG. 11, means that there are more Red Party supporters than Blue Party supporters in the community (see the key in Table 1). In this case, we say the community favors the Red Party. A white number in a black circle, such as second element set 508 in FIG. 11, means that there are more Blue Party supporters than Red Party supporters in the community. In this case, we say the community favors the Blue Party. The number itself is the margin (which may be in thousands of voters) by which the community supports one party over the other. For example, the black 3 in the white circle of second element set 508 means that there are 3000 more Red Party supporters than Blue Party supporters in the community. In this case, we say that the community's voter margin is “+3 Red.” The white 7 in the black circle of first element set 506 means that there are 7000 more Blue Party supporters than Red Party supporters in the community. In this case, we say that the community's voter margin is “+7 Blue.” A zero means that the community equally supports the two parties. In this case, we say that the community's voter margin is 0. In some examples, every community is designated as having the same total population.
• As discussed above, in this example there are a total of thirty six sector tiles 610. The seventeen sector tiles favoring the Red party are identical with respect to their voter margins to the seventeen sector tiles favoring the Blue party, and two sectors have a voter margin of 0. Hence, the overall voter margin in the state is 0; the same number of voters support each party statewide.
• In this example, the thirty six sector tiles 610—which remain in their initial positions as sectors 504 once placed for the game—are used as building blocks to form nine political districts that will cover the region 502. The nine districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray. Initially, no sector 504 belongs to any district. During the game, players use colored markers to assign communities to political districts. Each community eventually belongs to exactly one political district. Since nine districts will be created from thirty six sectors 504, at the end of the game the size of the average district—the number of sectors it has—will be four. However, the rules may permit variation in the size of a district, so some districts may be smaller or larger than others.
• The voter margin of a district depends on the voter margins of the sectors 504 that comprise it. The voter margin of a district equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district. A district's voter margin indicates which party has more voters in the district. For example, if a district has four communities with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district's voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district's voter margin is 0.
Scoreboard and Scoring Tokens
• During the game, the current voter margin of each district is indicated by the position and orientation of its scoring token 606 on the scoreboard 604. In particular, each district's scoring token 606 must always be placed so that (1) the value in the square it occupies plus (2) the number on the side of the scoring token that faces up equals the district's current voter margin. A scoring token may be cubed shaped, and may have its faces marked in the following manner: unmarked, 10, 20, 30, 40, 50. The side of a scoring token 606 that should be face up depends upon the range of the voter margin, and may correspond to unmarked: 0-12, 10:13-22, 20:23-32, 30:33-42, 40:43-52, 50:53-62. The scoring token 606 for a district may be placed on a square within its row 608 when a district's voter margin favors the (Blue, Red) Party respectively. For example, consider a moment in the game when three communities with voter margins “+5 Blue,” “+1 Red,” and “+4 Blue” have been assigned to the Green District. In this case, the Green District's voter margin is “+8 Blue” (=5+4−1), so the green scoring token should be placed on square “Green District Voter Margin=+8 Blue” with its unmarked side facing up. If a community with voter margin “+9 Blue” were added to this district, its new voter margin would be “+17 Blue” (=9+5+4−1), and the green scoring token would be moved to square “Green District Voter Margin=+7 Blue” with its “10” side facing up. Alternatively, if a community with voter margin “+9 Red” were added to this district, its new voter margin would be “+1 Red” (=9+1−5−4), and the green scoring token would be moved to square “+1 Red” with its unmarked side facing up.
Playing the Game
• Play may include the following three phases, although the second phase is optional.
• • 1. Build political districts
  • 2. Run an election (optional)
  • 3. Identify the winner
Phase 1: Build Political Districts Summary
• The first phase is the main phase of the game. During this phase, players may take turns assigning sectors 504 to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector 504 to a political district is accomplished by placing a home base marker 600 or expansion marker 602 on a vacant sector 504. Players may also reassign sectors from large districts to adjacent smaller districts to better equalize the district sizes. This is done by changing the color of the marker on a sector 504. At the end of this phase, there will be 9 non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—that cover the region 502.
• Each district evolves in the same way. Initially, it is formless. At some point, it is established when its home base marker 600 is placed on a vacant sector 504. (A vacant sector is a sector with no marker on it.) It is then expanded whenever one of its expansion markers 602 is placed on a vacant sector 504 that is adjacent to a sector 504 that already belongs to the district.
• The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There is no district size requirement. However, the rules encourage the creation of districts having four sectors 504.
• Importantly, all marker subsets 620 (consisting of the home base marker and expansion markers for a given color) and all sectors 504 are available to all players. No player “owns” any marker subset or sector 504. As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector 504 from that district to another district.
Details
• Note: During play, the voter margins of all sectors 504 are visible to both players. In FIGS. 15-20, however, the grid numbers 1-36 are shown instead of the sector voter margins, for ease of reference.
• Players may take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn (i.e. passing on a turn) is not allowed.
• All moves must be of type 1, 1A, 2, 2A, 3, or 3A (described below). Moves of type 1 and 1A establish a new district. These moves are in category E. Moves of type 2 and 2A expand an existing (i.e. already established) district. These moves are in category X. Moves of type 3 and 3A resize two adjacent districts. These moves are in category R. “A” means “alternate move.”
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than nine districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist.
• The six types of legal moves are as follows. FIGS. 15-20 illustrate different examples when certain move types are allowed by the rules. Explanations for the asterisked terms are provided at the end of these descriptions.
• • 1 Establish a new district by placing its home base marker 600 on a vacant sector 504. This move must meet two requirements. (a) The sector must be 2 or more (horizontal+vertical) steps away from each previously placed home base marker. The two sectors shown in (i), (ii), and (iii) in FIG. 18 are 1, 2, and 2 steps away from each other respectively. Sectors 20 and 29 in FIG. 12 are four steps away from each other. (b) There must be space to grow this district to a size of 4 connected* sectors.
    • The position shown in FIG. 15A illustrates moves of type 1. Here, placing a home base marker on sector 10, 11, or 17 is not allowed because it violates requirement (a). Also, placing a home base marker on sector 30, 35, or 36 is not allowed because it violates requirement (b). In this position there are only two possible moves of type 1: place a home base marker on sector 12 or sector 31.
  • 1A Establish a new district by placing its home base marker 600 on a vacant sector 504. This move must meet two requirements. (c) The sector must be in the largest open space on the board. An open space is a set of connected* vacant sectors. (d) The sector must be the farthest (in number of steps) from a previously placed home base marker (among the sectors satisfying requirement (c)).
    • The position shown in FIG. 15B illustrates moves of type 1A. Here, placing a home base marker on sector 3 or 17 is not allowed because it violates requirement (c). Also, placing a home base marker on sector 30 or 35 is not allowed because it violates requirement (d). In this position there are only three possible moves of type 1A: place a home base marker on sector 14, 19, or 36.
  • 2 Expand an existing district by placing one of its expansion markers 602 on a vacant sector 504. This move must meet three requirements. (e) The district must remain connected.* (f) The district's new size after this move—including the new sector and any sectors that are captured**—may not exceed 4 sectors. (g) No district may be trapped.***
    • The position shown in FIG. 16A illustrates moves of type 2. Here, placing an orange expansion marker 602 on sector 3 or 23 is not allowed because (e) is violated. Also, placing a yellow expansion marker 602 on sector 10 is not allowed because (f) is violated. Also, placing a purple expansion marker 602 on sector 7 or 13 is not allowed because sector(s) are captured and (f) is violated. Finally, placing a gray expansion marker 602 on sector 30 is not allowed because the Green District would be trapped and (g) would be violated.
  • 2A Expand an existing district by placing one of its expansion markers 602 on a vacant sector 504. This move must meet three requirements. (e) The district must remain connected.* (h) Only the smallest expandable district may be expanded. A district is expandable if there is at least one vacant sector adjacent to it. (i) No sectors may be captured.**
    • The position shown in FIG. 16B illustrates moves of type 2A. Here, placing a blue expansion marker on sector 6 is not allowed because (e) is violated. Also, placing an orange expansion marker on sector 35 is not allowed because (h) is violated. Finally, placing a blue expansion marker on sector 35 is not allowed because (i) would be violated. In this position there are only five possible moves of type 2A: place a brown expansion marker on sector 6; place a gray expansion marker on sector 6 or 24; or place a blue expansion marker on sector 24 or 36.
  • 3 Reassign a community from one district (say District X) to another (say District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet five requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (l) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected.* (n) The District X marker that is removed must be an expansion marker; it may not be a home base marker.
    • The position shown in FIG. 17A illustrates moves of type 3. Here, reassigning sector 13 to the Red District is not allowed because the Red District has not been established and (j) is violated. Reassigning sector 16 to the Gray District is not allowed because the Gray District is expandable and (k) is violated. Reassigning sector 9 to the Green District is not allowed because (l) is violated. Reassigning sector 23 or 35 to the Pink District is not allowed because (m) is violated. Reassigning sector 24 or 36 to the Pink District is not allowed because (n) is violated. In this position there are only two possible moves of type 3: reassign sector 16 to the Green District or reassign sector 28 to the Pink District.
  • 3A This move has the same requirements as move type 3 except that (n) is not required
    • The position shown in FIG. 17B illustrates moves of type 3A. In this position, six moves of type 3A are available: reassign sector 26 or 31 to the Red District; reassign sector 26 to the Purple District; and reassign sector 16, 23, or 27 to the Pink District.
• * See subsection entitled “Connectedness” below
• * See subsection entitled “Captured sectors” below
• ** See subsection entitled “Trapped districts” below
Connectedness
• Two sectors are adjacent—and connected—if and only if they share a common edge. For example, the two-sector area shown in (i) in FIG. 18 is connected, but the two-sector areas shown in (ii) and (iii) in FIG. 18 are NOT connected.
• In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X—a sequence of adjacent sectors that all belong to District X—connecting those two sectors.
Captured Sectors
• A set of connected, vacant sectors is captured if it is (i) surrounded by a single district or (ii) surrounded by the edge of the board on one side and a single district on the other side. In FIG. 19, sector 1 is captured by the Green District (consisting of sectors 2 and 7-8); sectors 5-6 are captured by the Blue District; sectors 31-36 are captured by the Red District; and sector 16 is captured by the Blue District. No other sectors are captured.
• A move of type 2 which captures exactly one sector is allowed if the district's new size—including the sector on which the marker is placed and the sector that is captured—is no greater than four sectors. All other moves that capture sectors are forbidden. For example, if sector 1 is vacant, it is permissible to add sector 2 to a district consisting of sectors 7-8. In this case, sector 1 is captured and the new district consists of sectors 1-2 and 7-8. However, if sectors 5-6 are vacant, adding sector 12 to a district consisting of sectors 4 and 10-11 is not allowed.
• A sector that is captured during a legal move of type 2 is immediately assigned to the district that has captured it. An expansion marker is immediately placed on this sector, and the scoreboard is updated appropriately.
Trapped Districts
• A district is trapped if (i) it (and the open spaces beside it) is either surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than four.
• In FIG. 20, the Gray District (consisting of sector 1) is trapped by the Green District (consisting of sectors 2 and 7-8) and the Pink and Orange Districts are trapped by the Blue District. The Yellow District is not trapped because it can still grow to a size of four sectors.
• A move of type 2 which traps a district is forbidden. For example, if the Gray District consists of sector 1 and the Green District consists of sectors 7-8, then an expansion of the Green District to sector 2 is not allowed. Also, placing a blue expansion marker on sector 15 to achieve the position in FIG. 20 is not allowed.
End of Phase 1
• Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into nine political districts that average four sectors each.
Phase 2: Run an Election Summary
• This optional phase of the game accounts for the surprises that can happen in real-world elections. Sometimes the candidate whose party has the majority of voters in a district is defeated by his/her opponent. This may happen if a candidate lacks charisma, public speaking skills, good looks, or other personal qualities or if the candidate takes unpopular stands on issues such as education, health care, the economy, infrastructure, foreign affairs, the environment, etc. In this phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice 618.
Details
• Each district is considered one at a time beginning with the Brown District.
• First, using the table below, the voter margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+8 Red” voter margin in the Yellow District converts to a 97% chance for the Red Party to win an election in the Yellow District.

• 	District
  	Voter	Winning
  	Margin	Likelihood

  	0	50%
  	+1	60%
  	+2	69%
  	+3	77%
  	+4	84%
  	+5	90%
  	+6	93%
  	+7	95%
  	+8	97%
  	+9	99%
  	+10 or more	100%

• Second, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The result shown on the black (white) die is the value of the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100.
• Third, the random number is compared to the winning percentage (e.g. 97 for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Red Party wins the Yellow District election if the random number is from 1-97, and the Blue Party wins the Yellow District election if the random number is from 98-100. If both parties have a 50% chance of winning a district, the Blue Party wins if the random number is from 1-50 and the Red Party wins if the random number is from 51-100. If a party has a 100% chance of winning a district, it automatically wins that district without a dice roll. After the winner of an election is identified, the scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district's portion of the scoreboard 604.
• The above procedure is repeated for each of the nine districts.
Phase 3: Identify the Winner
• In the game's final phase, the overall winner is identified.
• If phase 2 is played, the winner is the player whose party wins five or more district elections. If phase 2 is not played, players identify the party that controls each district, i.e. the party with more voters in each district. This is done by looking at the positions of the scoring tokens on the scoreboard. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie.
Example of Play
• FIGS. 21-30 provide an example of play. After the setup is finished, assume that the initial sector arrangement is as shown in FIG. 11. During play, sector grid numbers are not visible to the players. In FIGS. 21-29, however, sector grid numbers 1-36 are shown for ease of reference.
• During stage 1 of play, only moves of type 1, 2, and 3 are allowed. After 15 turns, assume the position in FIG. 21 is reached. An guide to the specific home base markers 600 and expansion markers 602 shown is provided in Table 1.
• The available moves of type 1 in this position are as follows:
• • Establish a new district on sector 1, 13, 15, 18, 21, 22, 24, 27, 34, or 36
• The available moves of type 2 in this position are as follows:
• • Expand Red District to sector 5
  • Expand Orange District to sector 19 or 31
  • Expand Yellow District to sector 17 or 18
  • Expand Blue District to sector 22, 23, 24, 27, or 36
  • Expand Purple District to sector 1, 3, 7, 9, or 14
  • Expand Pink District to sector 3, 5, 9, 15, 17, or 22
  • Expand Gray District to sector 14, 21, 27, 31, or 33
• No moves of type 3 are available in this position.
• Notes:
• • Sector 1 is captured and immediately added to the Purple District if the Purple District is expanded to sector 7.
  • An expansion of the Blue District to sector 34 or 35 is not allowed because it violates requirement (f).
  • An expansion of the Yellow District to sector 5 is not allowed because the Red District would be trapped and requirement (g) would be violated.
  • An expansion of the Gray District to sector 19 is not allowed because the Orange District would be trapped and requirement (g) would be violated.
• Eight moves later in the game, after a total of 23 moves, the new position is shown in FIG. 22. The game is still in stage 1 because a move of type 1 is available.
• The available moves of type 1 in this position are as follows:
• • Establish Green District on sector 21, 22, or 27
• The available moves of type 2 in this position are as follows:
• • Expand Brown District to sector 33 or 35
  • Expand Orange District to sector 19
• No moves of type 3 are available in this position.
• Notes:
• • Reassigning sector 4 from the Pink District to the Red District is not allowed because sector 4 is occupied by a home base marker (see requirement (n)).
  • Similarly, reassigning sector 11 from the Yellow to Red District is not allowed.
  • Reassigning sector 12 from the Yellow District to the Red District is not allowed because the Yellow District would not be connected (see requirement (m)).
• Four moves later, after a total of 27 moves, the new position is shown in FIG. 23. Here, (i) there is no way to make a move of type 1 that satisfies its criteria and (ii) fewer than nine districts have been established. Thus, stage 2 of play may begin. The next move must be of type 1A, 2, or 3.
• The available moves of type 1A in this position are as follows:
• • Establish Green District on sector 22
• The available moves of type 2 in this position are as follows:
• • Expand Orange District to sector 13
• No moves of type 3 are available in this position.
• Notes:
• • A move of type 1A requires that a new district be established on a sector that is in the largest open space on the board. Among the sectors satisfying this requirement, a sector that ties for being the most steps away from a previously placed home base marker must be selected. In the current position, the largest open space consists of three sectors: 17, 22, and 23. Among these sectors, only one—sector 22—is two or more steps away from all previously placed home base markers. So there is only one legal move of type 1A in this position.
• Two moves later, after a total of 29 moves, the new position is shown in FIG. 24. Here, all nine districts have been established, so we are in stage 3 of play. In stage 3, the next move must be of type 2 or 3A whenever a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A.
• The available moves of type 2 in this position are as follows:
• • Expand Orange District to sector 14
  • Expand Green District to sector 23
• The available moves of type 3A in this position are as follows:
• • Reassign sector 4 from Pink District to Red District
  • Reassign sector 11 from Yellow District to Red District
• Notes:
• • The Red District is the only district that is not expandable. Thus, it is the only district that could possibly “steal” a community from another district (see requirement (k)).
• Three moves later, after a total of 32 moves, the new position is shown in FIG. 25. In this position, no move of type 2 exists. Thus, the next move must be of type 2A or 3A.
• The available moves of type 2A in this position are as follows:
• • Expand Brown District to sector 35
  • Expand Blue District to sector 35
  • Expand Purple District to sector 3 or 9
  • Expand Pink District to sector 3 or 9
• The available moves of type 3A in this position are as follows:
• • Reassign sector 4 from Pink District to Red District
  • Reassign sector 11 from Yellow District to Red District
• Notes:
• • Four districts—Brown, Blue, Purple, Pink—currently tie for being the smallest expandable district (see requirement (h)).
• Two moves later. After a total of 34 moves, the new position is shown in FIG. 26. In this position, a move of type 2 is available. Thus, the next move must be of type 2 or 3A.
• The available moves of type 2 in this position are as follows:
• • Expand Red District to sector 3
• The available moves of type 3A in this position are as follows:
• • Reassign sector 9 from Purple District to Pink District
• One move later, after a total of 35 moves, the new position is shown in FIG. 27. Here, no move of type 2 exists. Thus, the next move must be of type 2A or 3A.
• The available moves of type 2A in this position are as follows:
• • Expand Brown District to sector 35
  • Expand Blue District to sector 35
• The available moves of type 3A in this position are as follows:
• • Reassign sector 9 from Purple District to Pink District
• One move later, after a total of 36 moves, the new position is shown in FIG. 28. There are no vacant sectors, so all future moves will be of type 3A. Play continues until no such moves exist.
• The available moves in this position are as follows:
• • Reassign sector 21 from Brown District to Green District
  • Reassign sector 21 from Brown District to Pink District
  • Reassign sector 9 from Purple District to Pink District
• One move later, after a total of 37 moves, the new position is shown in FIG. 29. No legal moves exist in this position, so phase 1 of play concludes. The districts that have been formed are now final.
• The final position at the end of phase 1 is shown in FIG. 30, with the districts 510-526 marked. The markers played and the community voter margins are shown.
• The final district voter margins are shown on the scoreboard (see table below).
• If phase 2 is not played, the game immediately ends, and the Blue Party wins by a score of 5 districts to 3 districts (with one tied district).

• 		No.	Voter
  	District	Communities	Margin

  	1. Brown	4	0	(=6 + 3 − 5 − 4)
  	2. Red	4	+7 Blue	(=0 + 8 + 6 − 7)
  	3. Orange	4	+3 Blue	(=5 + 3 − 3 − 2)
  	4. Yellow	4	+8 Red	(=5 + 9 − 2 − 4)
  	5. Green	3	+5 Red	(=6 + 7 − 8)
  	6. Blue	4	+2 Blue	(=9 + 2 − 1 − 8)
  	7. Purple	5	+9 Red	(=1 + 2 + 8 − 1 − 1)
  	8. Pink	4	+1 Blue	(=7 + 4 − 6 − 4)
  	9. Gray	4	+9 Blue	(=7 + 0 + 5 − 3)

• If phase 2 is played, dice 618 are rolled to determine the winning party in each district. In the game at hand, the final voter margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +7 Blue, +3 Blue, +8 Red, +5 Red, +2 Blue, +9 Red, +1 Blue, +9 Blue). Using a preceding table, these margins translate to winning likelihoods of (50%, 95%, 77%, 97%, 90%, 69%, 99%, 60%, 99%) for the parties with the majority of voters in these districts respectively. Note that each party has a 50% chance of winning the Brown District, and no party automatically wins a district with 100% probability.
• Dice 618 are then thrown to determine the election results. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in five out of nine districts. The Red Party wins the game by a score of 5 districts to 4 districts.

• 		Voter	Winning	Dice	Election
  	District	Margin	Likelihood	Roll	Result

  Brown

  	0	50% for Blue	68	Red Wins
  	Red	+7 Blue	95% for Blue	95	Blue Wins
  	Orange	+3 Blue	77% for Blue	4	Blue Wins
  	Yellow	+8 Red	97% for Red	25	Red Wins
  	Green	+5 Red	90% for Red	41	Red Wins
  	Blue	+2 Blue	69% for Blue	13	Blue Wins
  	Purple	+9 Red	99% for Red	92	Red Wins
  	Pink	+1 Blue	60% for Blue	61	Red Wins
  	Gray	+9 Blue	99% for Blue	30	Blue Wins

Rules for a Symmetric Game
• A starting region with a large connected portion of high-numbered sectors favoring the Blue Party but no large connected portion of high-numbered sectors favoring the Red Party is biased in favor of the Red Party. In such a setting, the player representing the Red Party will more easily be able to concentrate or “pack” the voting power of the opposing party into a small number of districts than the player representing the Blue Party. Thus, the Red Party is more likely to win the game.
• The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game. This is particularly important in a tournament setting.
• A symmetric game has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
• • 1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the region (see FIG. 12). In a counter-symmetric sector arrangement the two sectors comprising every pair of diametrically-opposed sectors—every two sectors that are on exact opposite sides of the state—have the same voter margins but they favor different parties. This arrangement guarantees that the starting map is unbiased, favoring neither party.
    • FIG. 31 shows a counter-symmetric sector arrangement. Three portions of the region 502 are shown: the central square 622; the inner ring 624 which encircles it; and the outer ring 626 which encircles the inner ring. Bold lines distinguish these three portions of the region. Note that the sectors on opposite sides of the central square have the same number but opposite voter margins. Sector 15 is “+6 Red” whereas sector 22 is “+6 Blue,” and sector 16 is “+7 Blue” whereas sector 21 is “+7 Red” (see FIG. 12 for the sector number key). The sectors on opposite sides of the inner ring 624 also have the same number but opposite voter margins. The same holds true for the sectors in the outer ring 626. This map's symmetry gives each player a fair chance of winning the game.
    • A random counter-symmetric initial sector arrangement can be efficiently created using the “Symmetric Game Setup Table” 628 shown in FIG. 32 and any thirty six expansion markers. The procedure works as follows. First, all markers are removed from the Symmetric Game Setup Table. The thirty-six sector tiles are then mixed and organized face down into a single deck. Sector tiles are drawn from the deck one at a time. When a sector tile is drawn, players look at the portion of the Symmetric Game Setup Table 628 that matches the sector's voter margin and color (e.g. “+5” and “Blue”). If the total number of markers in this portion of the table is greater than or equal to the total number of markers in the portion of the table with the same voter margin but opposite color, the sector tile is placed face up in the first unoccupied location according to the sector sequence in FIG. 12. Otherwise the sector tile is placed face up in an unoccupied location that is diametrically opposed to where an opposing sector tile (with the opposite voter margin) has already been placed. Then a marker is placed on a dot in the Symmetric Game Setup Table that matches the sector's voter margin and color. This continues until all 36 sector tiles are drawn and placed face up.
  • 2. During the second and third moves of phase 1 (the 1^st move made by the player who goes second, and the 2^nd move made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which a marker has already been placed.
  • 3. During phase 1, a move (of any type) that creates a district that (a) has size four and (b) coincides with the central square is never allowed.
Tournament Play
• This example of the present technology a game of pure skill if (A) players decide who plays first prior to the start of the game, (B) a symmetric game is played, and (C) phase 2 of the game is skipped. This form of the game, like international chess and the Japanese game go, is highly suited to tournament play. Unlike chess and go, the initial board position in this game is always different, so every game has a unique opening.
Handicap Play
• This game is suited to handicap play. If the players' skill levels differ, the playing field can be leveled by changing the voter margin of one or more sectors. For example, if the stronger player represents the Blue Party, the players may agree, before any sector tiles 610 are placed, to change the voter margin of the first “+8 Blue” sector tile that is placed from “+8 Blue” to 0. Alternatively, the weaker player may be allowed to make more than 50% of the moves—for example 5 of every 9 moves.
Game Alternative #1: Form Seven Districts (Each of Size 5)
• In one variation of strategy game 500, seven districts are formed instead of nine districts. In this variation, only seven marker subsets 620, representing seven district colors, are used, and the average size of a district at the end of the game is about five sectors 504. At the end of the game, the region will be divided into seven political districts. This variation of the game may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 are slightly different as described below:
• • A move of type 1 must meet two requirements. (a) The sector on which the home base marker is placed may not be in the ring surrounding any previously placed home base marker. (b) There must be space to grow the new district to a size of 5 connected sectors. In FIG. 12, the ring surrounding sector 1 consists of sectors 2 and 7-8. The ring surrounding sector 10 consists of sectors 3-5, 9, 11, and 15-17. The ring surrounding sector 5 consists of sectors 4, 6, and 10-12.
  • A move of type 1A must meet two requirements. (c) The sector on which the home base marker is placed must be in the largest open space on the board. (d) Among the sectors satisfying the requirement c, the sector must be one that is not in a ring surrounding a previously placed home base marker. If no such sector exists, the sector must satisfy requirement c and be the farthest (in number of horizontal+vertical steps) from a previously placed home base marker.
  • A move of type 2 must meet three requirements. (e) The district must remain connected. (f) The district's new size—including the new sector and any captured sectors—must not be greater than 5 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is either surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 5.
Example 4
• Strategy games of Example 4 have a taxonomic code A/G/S2/P/2/12/U/2/EXR. They are analog, multi-player games with 72 square sectors and a political focus in which two types of elements—namely two political parties—are present and twelve districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• One example is strategy game 700 as shown in FIGS. 33-35, which is a larger version of the game described in Example 3. Strategy game 700 includes a region 702, which as illustrated is set up as an 8×9 grid divided into sector placeholders 718 numbered 1-72. Sector tiles 716 are provided, which may be shuffled or otherwise reorganized, and laid out on the region with one sector tile 716 per sector placeholder 718 to form 72 sectors 704 in the region 702 at the start of a game, as shown in FIG. 34. The set of elements 724 for each sector 704 is provided on each sector tile 716.
Synopsis
• Strategy game 700 is very similar to strategy game 500 described in Example 3. The main differences are as follows. First, in strategy game 700, there are 72 sectors—exactly twice as many sectors of each kind as in strategy game 500. Second, at the start of strategy game 700 the sector tiles 716 are randomly placed in a 9×8 rectangular arrangement within region 702 to form sectors 704. Third, twelve districts—the average size of which at the end of the game may be six sectors 704—will be formed. A set of markers 726 containing a total of 12 marker subsets 710 may be used, each marker subset 710 representing a district and consisting of one home base marker 712 and eighteen expansion markers 714. As shown there are nine rectangular home base markers (one of each for Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) and 162 square expansion markers (18 of each of the same colors as the rectangular home base markers), and then three additional home base markers that are diamond shaped (one each for the Brown2, Red2, and Orange2 Districts) and three additional sets of expansion markers that are triangular in shape (18 of each of the same colors as the diamond shaped home base markers). Each district is distinguishable by the color and/or shape of the markers used to form it. Fourth, the precise rules for making moves of types 1, 1A, 2, and 2A are slightly different in this game to encourage most districts to have a size of six sectors 704 at the end of the strategy game 700.
• In the illustrated example, strategy game 700 includes two scoreboards, a first scoreboard 706 and a second scoreboard 708. Each scoreboard has a plurality of rows 722 and looks like FIG. 13. The total number of rows in both scoreboards is at least 12, and thus one row may be used to track the voter margin of each district. There are also twelve scoring tokens 720, one for each district. In other examples of strategy game 700, there may only be one scoreboard, which would have one row 722 for each of the twelve districts to be formed during the game.
• As illustrated, FIG. 34 shows the region 702 at the start of strategy game 700, and FIG. 35 shown one possible solution at the end of game play. In FIG. 35, the home base markers 712 and expansion markers 714 have been placed in a manner that establishes twelve districts 728-750.
• The table below shows the final result of this game. The Blue Party wins this example game by a score of 6 districts to 5 districts (with one tied district).

• 		Voter
  	District	Margin
  	Brown	+2 Blue
  	Red	+13 Blue
  	Orange	+5 Red
  	Yellow	+4 Red
  	Green	+9 Red
  	Blue	+2 Blue
  	Purple	+5 Blue
  	Pink	+4 Blue
  	Gray	+2 Red
  	Brown2
  	0
  	Red2	+4 Blue
  	Orange2	+10 Red

Playing the Game
• Strategy game 700 may have the same three phases as, and may be played in a manner that is nearly identical to, Example 3.
Phase 1: Build Political Districts
• During a player's turn, he/she (A) makes one move and (B) records the move on the appropriate scoreboard 706 or 708. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 80 moves—40 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play may be divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist. Forfeiting a turn (i.e. passing on a turn) is not allowed.
• The rules may provide six types of legal moves, such as those listed below. The terms “captured” and “connected” have the same meaning as described in Example 3. Moves of type 3 and 3A are identical to Example 3.
• • 1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (a) The sector may not be in the ring surrounding any previously placed home base marker. In FIG. 33, the ring surrounding sector 1 consists of sectors 2 and 9-10. The ring surrounding sector 10 consists of sectors 1-3, 9, 11, and 17-19. The ring surrounding sector 5 consists of sectors 4, 6, and 12-14. (b) There must be space to grow this district to a size of 6 connected sectors.
  • 1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (c) The sector must be in the largest open space on the board. An open space is a set of connected vacant sectors. (d) Among the sectors satisfying requirement c, the sector must be one that is not in a ring surrounding a previously placed home base marker. If no such sector exists, the sector must satisfy requirement c and be the farthest (in number of horizontal+vertical steps) from a previously placed home base marker. Sectors 2 and 21 in FIG. 33 are five steps away from each other.
  • 2 Expand an established district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e) The district must remain connected. (f) The district's new size—including this new sector and any captured sectors—must not be greater than 6 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is either completely surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 6.
  • 2A Expand an established district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (e) The district must remain connected. (h) Only the smallest expandable district may be expanded. A district is expandable if there is at least one vacant sector adjacent to it. (i) No sectors may be captured. (o) The district's new size must not be greater than 19 sectors. This last requirement relates to limited marker quantities.
  • 3 Reassign a community from one district (say District X) to another (say District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet five requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (l) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected after this move. (n) The District X marker that is removed must be an expansion marker; it may not be a home base marker.
  • 3A This move has the same requirements as move type 3 except that (n) is not required.
End of Phase 1
• Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into 12 political districts that average 6 sectors each. FIG. 35 shows a possible position at the end of phase 1.
Phase 2: Run an Election
• This phase of the game is nearly identical to Example 3 except that a different table (shown below) is used to convert a district's voter margin into the probability that the party with more voters in the district wins an election in the district.

• 	District
  	Voter	Winning
  	Margin	Likelihood

  	0	50%
  	+1	59%
  	+2	67%
  	+3	74%
  	+4	80%
  	+5	85%
  	+6	89%
  	+7	92%
  	+8	94%
  	+9	96%
  	+10	98%
  	+11	99%
  	+12 or more	100%

Phase 3: Identify the Winner
• This phase of the game is exactly the same as in Example 3. If phase 2 is played, the winner is the player whose party wins seven or more district elections. If each party wins six district elections, the result is a tie. If phase 2 is not played, players look at the scoring tokens on the scoreboard to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie.
• Game Alternative #1: Form 18 Districts (with an Average Size of 4 Sectors)
• One variation of strategy game 700 includes the formation of 18 districts—instead of 12—in the same 9×8 region 702. In this variation, exactly twice as many home base, expansion, and scoring tokens are used compared to Example 3. The size of the average district at the end of this variation of the game is 4 sectors 704. This variation may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 may be slightly different as described below:
• • A move of type 1 must meet two requirements. (a) The sector on which a home base marker is placed must be at least 2 horizontal+vertical steps away from all previously placed home base markers. (b) There must be space to grow the new district to a size of 4 connected sectors. In FIG. 33, sectors 20 and 54 are six steps away from each other.
  • A move of type 1A must meet two requirements. (c) The sector on which a home base marker is placed must be in the largest open space on the board. (d) Among the sectors satisfying the requirement c, the sector must tie for being the farthest (in number of horizontal+vertical steps) from a home base marker.
  • A move of type 2 must meet three requirements. (e) The district must remain connected. (f) The district's new size—including the new sector and any captured sectors—must not be greater than 4 sectors. (g) No district may be trapped. A district is trapped if (i) it (and the open spaces beside it) is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than 4.
Rules for a Symmetric Game
• As with strategy game 500, strategy game 700 can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game.
• When strategy game 700 is played as a symmetric game, there are two additional rules that may be used compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rule 2 reduces the possibility of a symmetric position during play. The two rules are as follows.
• • 1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the state. This arrangement guarantees that the starting map is unbiased, favoring neither party. A procedure for doing this may be similar or nearly identical to that described in the subsection “Rules for a symmetric game” in the description of Example 3.
  • 2. During the 2^nd, 3^rd, 4^th, and 5^th moves of phase 1 (i.e. the 1^st and 2^nd moves made by the player who goes second, and the 2^nd and 3^rd moves made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which another marker has already been placed.
Example 5
• Strategy games of Example 5 have a taxonomic code A/G/S3/P/2/15/U/2/EXR. They are analog, multi-player games with 90 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• Strategy games of Example 5 are is very similar to the games described in Examples 3-4. In one example, two players—Red and Blue—vie for political control of a 9×10 rectangular state by competitively creating 15 political districts (whose average size is 6) out of 90 square communities. The game can be played with any 90 sectors in which the sets of red and blue sectors are identical—for example 5 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (80 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 4 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves—50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• The six types of moves allowed in the rules for games of this type are summarized below.
• • 1 Establish a district on a sector that is not in the ring surrounding any other home base marker. There must be space to grow this district to size 6 or more.
  • 1A Establish a district on a sector that is not in the ring surrounding any other home base marker (among the sectors in the largest open space on the board). If this is not possible, establish a district on a sector that is farthest (in number of steps) from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 6 or less, and (g) no district is trapped.
  • 2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less.
  • 3 (Same as in Example 3).
  • 3A (Same as in Example 3).
Example 6
• Strategy games of Example 6 are larger versions of the games described in Examples 3-5. These games have a taxonomic code A/G/S4/P/2/15/U/2/EXR. They are analog, multi-player games with 121 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• In at least one example, two players—Red and Blue—vie for political control of an 11×11 square region by competitively creating 15 political districts (whose average size is just above 8) out of 121 square sectors, each of which represents a community. The game can be played with any 121 sectors in which the sets of red and blue sectors are identical—for example 7 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (112 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 140 moves—70 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• The six types of legal moves are summarized below.
• • 1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 8 or more.
  • 1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g) no district is trapped.
  • 2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less.
  • 3 (Same as in Examples 3-5)
  • 3A (Same as in Examples 3-5)
Example 7
• Strategy games of Example 7 are larger versions of the games described in Examples 3-6. They have a taxonomic code A/G/S5/P/2/21/U/2/EXR. They are analog, multi-player games with 169 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. Games of this type proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• In one example, two players—Red and Blue—vie for political control of a 13×13 region by competitively creating 21 political districts (whose average size is just above 8) out of 169 square sectors. The game can be played with any 169 sectors in which the sets of red and blue sectors are identical—for example 10 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (160 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 180 moves—90 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21^st district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• The six types of legal moves are summarized below.
• • 1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 8 or more.
  • 1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g) no district is trapped.
  • 2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 19 or less
  • 3 (Same as in Examples 3-6)
  • 3A (Same as in Examples 3-6)
Example 8
• This is a larger version of the games described in Examples 3-7. This game has taxonomic code A/G/S6/P/2/21/U/2/EXR. It is an analog, multi-player game with 210 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• This game is very similar to Examples 3-7. In this game, two players—Red and Blue—vie for political control of a 15×14 rectangular state by competitively creating 21 political districts (whose average size is 10) out of 210 square communities. The game can be played with any 210 sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (192 sector tiles); 6 each of sector tiles “+9 Red” and “+9 Blue” (12 sector tiles); and 6 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21^st district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• The six types of legal moves are summarized below.
• • 1 Establish a district on a sector that is at least 3 (horizontal+vertical) steps away from all other home base markers. There must be space to grow this district to size 10 or more.
  • 1A Establish a district on a sector that ties for being the farthest (in number of horizontal+vertical steps) from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e) it remains connected, (f) its new size (including any captured sectors) is 10 or less, and (g) no district is trapped.
  • 2A Expand the smallest expandable district so (e) it remains connected, (i) no sectors are captured, and (o) its new size is 37 or less.
  • 3 (Same as in Examples 3-7)
  • 3A (Same as in Examples 3-7)
Example 9
• Strategy games of Example 9 have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. The game paradigm is simultaneous independent play, in which each player makes one move per turn in sequential turns, independently of the other players. Any number of players—two or more—may play.
Game Summary
• In one example, illustrated in FIGS. 36-39B as strategy game 800, players compete, optionally under time constraints, to see who can best create the nine political districts of a square region 802, which represents a state. The region 802 consists of 36 square sector placeholders 806 that are formed into 36 sectors 804 at the start of play, each sector 804 representing a community. The region 802 has an American-style, two-party political system in which one person is elected to represent each political district. At the outset, the districts are formless and the players know the political status of each community (i.e. which party its citizens favor and by how much). During the first phase of the game, players simultaneously and independently work on identical copies of the region 802 to create political districts that achieve the pre-defined goal of maximizing the political advantage of the Red Party. During the (optional) second phase of the game, players simultaneously and independently work on identical copies of the same map to create political districts that achieve the pre-defined goal of maximizing the political advantage of the Blue Party. The winner is the player who does the best job of achieving the pre-defined goals during the game.
Components
• FIGS. 36-38 show the game components that may be used to play strategy game 800. A timer 814, which may be digital or analog, may be used in examples of strategy game 800 that are played under time constraints. In addition, each player should have a copy of the same game set which contains the following items:
• • A region 802
  • 36 sector tiles 820, which may have the same element sets as the sector tiles in Example 3
  • A set of expansion markers 822 divided into nine marker subsets consisting of a plurality of expansion markers 808. In this instance, 76 expansion markers in the following amounts and colors: 8 Brown, 10 Red, 8 Orange, 8 Yellow, 8 Green, 10 Blue, 8 Purple, 8 Pink, 8 Gray
  • 9 scoring tokens 812 (one for each of the nine districts to be formed)
  • A scoreboard 810, which may be identical to scoreboard 604
  • A game board 818 (FIG. 37), which may have a duplicate region 824 divided into 36 duplicate sector placeholders 826. The game board 818 may also have a plurality of rows 828 configured to allow a player to track aspects of districts formed on the game board 818.
    Setup (about 10 Minutes)
• The players decide (yes or no) if phase 2 of the game will be played, and they agree upon a time limit for each phase of the game. The “yes” option with a 10-minute time limit is recommended. (Such a game lasts about 50 minutes.)
• One player may be selected as the leader. All players except the leader may organize their 36 sector tiles 820 into 19 face-up piles—one pile for each number+color combination—so that specific sector numbers and colors can be quickly located. The leader may spread their sector tiles 820 out face down, mix them, and organize them face down into a single deck. The leader may then draw the 36 sector tiles 820 from the deck one at a time and place them face up with one on each sector placeholder 806 of the region 802 to form sectors 804. Each time a sector is drawn, the leader may announce its (a) position in the sequence, (b) color, and (c) number—for example “Sector 1: Red 4,” “Sector 2: Blue 2,” “Sector 3: Zero,” etc.—so that every other player may find the same sector tile 820 from his/her game set and place it in the same location in his/her region 802. When this process ends, each player has a copy of the leader's sector arrangement in his/her region 802. One possible arrangement of sectors 804 in the region 802 is identical to that shown in FIG. 11.
Scoreboard and Scoring Tokens
• Each player may use his/her scoreboard 810 and scoring tokens 812 to keep track of the voter margins of the districts that he/she creates during play.
• At the end of each phase of the game, the leader or other players may visit each player's playing area to ensure that the positions and orientations of his/her scoring tokens properly show the voter margins of the districts that he/she has created. Each district's scoring token 812 should be placed so that (1) the value in the square it occupies on the scoreboard plus (2) the number on the side of the scoring token that faces up equals the district's voter margin. When the scoring tokens 812 are cubes—having sides that are unmarked and marked 10, 20, 30, 40, 50 respectively—the (unmarked, 10, 20, 30, 40, 50) side of a scoring token should face up if the voter margin of its district is in the range (0-12, 13-22, 23-32, 33-42, 43-52, 53-62) respectively. The scoring token 812 is placed on the scoreboard to reflect the voter margin in each district. For example, if Player 1's Green District contains sectors with voter margins “+6 Blue,” “+1 Red,” “+9 Blue,” and “+7 Blue,” then the Green District's voter margin is “+21 Blue” (=6+9+7−1) and Player 1's green scoring token should be placed on the square “Green District Voter Margin=+11 Blue” with its “10” side facing up. If Player 2's Gray District contains sectors with voter margins “+3 Blue,” “+1 Red,” “+9 Red,” and “+5 Blue,” then the Gray District's voter margin is “+2 Red” (=1+9−3−5) and Player 2's gray scoring token should be placed on the square “Gray District Voter Margin=+2 Red” with its unmarked side facing up.
Playing the Game
• Play may consist of the following two phases. The second phase is optional.
• • 1. Build political districts that maximize the Red Party's advantage
  • 2. Build political districts that maximize the Blue Party's advantage
    Phase 1: Build Political Districts that Maximize the Red Party's Advantage
• The timer 814 is set to the time limit agreed upon by the players, if time limits are being used. Play begins when the timer starts or the players agree to begin.
• During the first phase of the game, each player independently uses his/her colored markers to form 9 political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—on his/her 36 sectors 804. Each district is formed by placing four markers, one per turn, that match the district color on four adjacent sectors. Each player's main goal in this phase is to create 9 political districts—i.e. a district plan—in which the Red Party controls as many districts as possible. A party controls a district if it has the majority—strictly more than half—of the voters in a district. Each player's secondary goal is to make the “voter margin in the district that the Red Party controls by the least amount” as high as possible.
• The rules may require that each player's district plan must satisfy the following two requirements:
• • A. Each sector must belong to exactly one political district. That is, there must be exactly one expansion marker 808 on each sector 804.
  • B. Each political district must consist of four connected sectors 804.
• Each player is free to use his/her scoreboard, game board, and markers as desired. It is recommended that each player (a) use the expansion markers 808 beside his/her 36 sectors 804 and scoreboard to create and evaluate potential district plans and (b) use the duplicate region 824 on his/her game board 818 to store the best district plan that he/she has found. At the end of this phase, each player's final district plan must be displayed by a set of 36 markers (four per color) that are placed either on his/her region 802 or on the duplicate region 824 on his/her game board 818.
• When play concludes, the final district plan made by each player is scored. The scoring of each player's final district plan is done by (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following values:
• • 1. The number of districts controlled by the Red Party.
  • 2. The lowest voter margin in the districts controlled by the Red Party (“Lowest Voter Margin in Red Districts”).
• If a player's district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 1 and 2 respectively.
Illustrative Example
• One example of the result of phase 1 of a two-player version of game 800 is shown in FIGS. 38A and 38B. Bold lines around the districts created by the expansion markers 808 indicate the final district plans in phase 1 for each player.
• At this point, Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+3 Red, +2 Red, +10 Red, +18 Blue, +16 Blue, +12 Red, +5 Blue, +4 Red, +8 Red). The Red Party controls 6 districts, and the lowest voter margin in those six districts is “+2 Red.” Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+2 Red, +7 Blue, +7 Red, +2 Red, +3 Red, +3 Red, +5 Red, +19 Blue, +4 Red). The Red Party controls 7 districts, and the lowest voter margin in those seven districts is “+2 Red.” Overall, Player 2 has done better in this phase of the game because his/her district plan gives the Red Party control of more districts than Player 1's district plan. Each player uses two red markers to mark his/her scores for phase 1 in the first two rows 828 of his/her game board 818 as shown below.

• 	Game Board Item	Player	1	Player 2

  	1. No. Districts Controlled by Red Party	6	7
  	2. Lowest Voter Margin in Red Districts	2	2

  
  Phase 2: Build Political Districts that Maximize the Blue Party's Advantage
• At the end of phase 1, players remove all markers from their scoreboards and game boards, except the two red markers used to mark their final scores for phase 1 on their game boards. The timer, if used, is then set to the time limit agreed upon by the players. Play of phase 2 is then started.
• During phase 2, play proceeds exactly as in phase 1 except that now each player's (i) main goal is to create a district plan in which the Blue Party controls as many districts as possible and (ii) secondary goal is to make the “voter margin in the district that the Blue Party controls by the least” as high as possible. The rules for making districts are just as in phase 1.
• When the timer goes off, all players cease their activities, disengage from their playing areas, and assemble as a group in the middle of the room. Working as a team, the players together (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player's final district plan:
• • 3. The number of districts controlled by the Blue Party.
  • 4. The lowest voter margin in the districts controlled by the Blue Party (“Lowest Voter Margin in Blue Districts”).
• If a player's district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 3 and 4 respectively.
• The above quantities are remembered by placing two blue markers on the appropriate squares in rows 3 and 4 of each player's game board.
Illustrative Example
• One example of the result of phase 2 of the two-player version of game 800 is shown in FIGS. 39A and 39B. Bold lines around the districts created by the expansion markers 808 indicate the final district plans in phase 2 for each player.
• At this point, Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+5 Blue, +1 Blue, +10 Blue, +2 Blue, +1 Blue, +6 Red, +5 Blue, +4 Blue, +22 Red). The Blue Party controls 7 districts, and the lowest voter margin in those seven districts is “+1 Blue.” Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (“+10 Red,” “+15 Blue,” “+6 Blue,” “+3 Blue,” “+25 Red,” “+3 Blue,” 0, “+3 Blue,” “+5 Blue”). The Blue Party controls 6 districts, and the lowest voter margin in those six districts is “+3 Blue.” Overall, Player 1 has done better in this phase of the game because his/her district plan gives the Blue Party control of more districts than Player 2's district plan. Each player uses two blue markers to mark his/her scores for phase 2 in rows 3-4 of his/her game board 818 as shown below.

• 	Game Board Item	Player	1	Player 2

  	3. No. Districts Controlled by Blue Party	7	6
  	4. Lowest Voter Margin in Blue Districts	1	3

Identifying the Winner(s)
• The winner of the game is identified, by process of elimination, by looking at the markers in the rows 828 of each player's game board 818. These markers may show the scores for the following items:
• • 1. Number of districts controlled by the Red Party in phase 1
  • 2. Lowest voter margin in the districts controlled by the Red Party in phase 1
  • 3. Number of districts controlled by the Blue Party in phase 2 (if phase 2 played)
  • 4. Lowest voter margin in the districts controlled by the Blue Party in phase 2 (if phase 2 played)
• If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game.
• If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game. In the preceding illustrative example, the players' scores for items 1+3 have the same sum, so the sum of the scores for items 2+4 is the tiebreaker. The sum of Player 1's scores for items 2+4 is 3. The sum of Player 2's scores for items 2+4 is 5, so Player 2 wins.
Example 10
• Strategy games of Example 10 are similar to those of Example 9 and have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. These games proceed according to simultaneous independent play, so any number of players may play.
• In one example, the game is set up exactly as described in Example 9, but the pre-defined goal of each player is to create the most balanced set of political districts. In particular, each player's pre-defined goal may be to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player's district plan must satisfy requirements A-B (see description of Example 9).
• At the end of play, each player's final district plan must be displayed and scored. Scoring may include (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following for each player's final district plan:
• • 1. The number of districts controlled by the Red Party.
  • 2. The lowest voter margin in the districts controlled by the Red Party.
  • 3. The number of districts controlled by the Blue Party.
  • 4. The lowest voter margin in the districts controlled by the Blue Party.
  • 5. The magnitude of the difference between items 1 and 3 above (the “Red-Blue Control Differential”).
  • 6. The magnitude of the difference between items 2 and 4 above (the “Lowest Voter Margin Differential”).
  • 7. The number of tied districts (with a voter margin of 0).
• The above items may be remembered by placing two red, two blue, and three gray markers in appropriate places on the rows of each player's game board 828.
Identifying the Winner(s)
• The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 9), all players lose.
Example of Play
• FIGS. 40A and 40B illustrate one possible conclusion of a strategy game of Example 10, with Player 1's final district plan shown in FIG. 40A and Player 2's final district plan shown in FIG. 40B. Bold lines indicate the districts formed by the layout of expansion markers 808 for each player.
• Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+19 Blue, +5 Blue, 0, +2 Blue, +12 Red, +8 Blue, +4 Red, +7 Red, +11 Red). In player 1's district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 4; the lowest voter margin in the districts controlled by the Blue Party is 2; and one district is tied. Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +3 Blue, +15 Blue, +14 Blue, +3 Red, +25 Red, +15 Blue, +9 Red, +10 Red). In player 2's district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 3; the lowest voter margin in the districts controlled by the Blue Party is 3; and one district is tied.
• The scoring of items 1-7 above takes place in rows 1-7 of the game board and is summarized in the table below. Both players' district plans satisfy requirements A-B, and the players have the same score for item 5, so the score for item 6 is the tiebreaker. Player 2 has a lower score for item 6, so Player 2 wins the game.

• 		Player 1	Player 2
  	Item	Score	Score

  Item

  1	4	4
  	Item 2	4	3
  	Item 3	4	4
  	Item 4	2	3
  	Item 5	0	0
  	Item 6	2	0
  	Item 7	1	1

Example 11
• Strategy games of Example 11 are generally for 2-4 players and are similar to games of Example 3. They have taxonomic code A/G/S1/P/4/9/U/2-4/EXR. They are analog, multi-player games with 36 square sectors in which four types of elements—such as four political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play, and moves in categories “E,” “X,” and “R” are allowed.
Game Summary
• FIGS. 41-43 illustrate components of a strategy game 900. Game components not shown in FIGS. 41-43 may be identical to those shown in FIG. 12. In this game, 2-4 players representing opposing political parties (Red Lightning Bolts, Orange Suns, Green Diamonds, and Blue Moons) vie for political control of the region 902, which may represent a Martian colony, by competitively creating nine political districts out of 36 square sectors 904 in alternating turn-based fashion. At the outset, the districts are formless and the players know how many voters support each party in each community. During the first phase of the game, players build the political districts by assigning communities to political districts one community at a time. They may also reassign communities from large districts to adjacent smaller districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. In the game's final phase, the parties that control the districts are identified, and 3 (1) points are awarded to a party that has sole (joint) control of a district. The player whose party has more points than any other player's party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win.
• FIG. 41 shows examples of two sector tiles 910 that may be used in this game. Each sector tile 910 represents a community and has a set of elements 912 including four icons in its center. Each icon represents an element, such as one person, or a set number of people, who support a political party. There are four non-limiting examples of icons shown, though any suitable icons may be used—a lightning bolt (red), sun (orange), diamond (green), and moon (blue)—corresponding to the four political parties. The number of identical icons represents the quantity 914 of the particular element. For example, the community represented by the first sector tile 910 on the left of FIG. 41 contains four people each of whom supports a different party. The community represented by the second sector tile 910 on the right of FIG. 41 contains two people who support the Red Lightning Bolts and two people who support the Green Diamonds.
• FIGS. 42A and 42B provide a visual summary of one possible version of strategy game 900. FIG. 42A shows one possible starting position. FIG. 42B shows one possible final position for the game, with home base markers 906 and expansion markers 908 forming nine districts (indicated by bold lines). The table below shows the total number of voters supporting each party in each district of FIG. 42B at the end of the game. A superscript W indicates that a party wins a district outright. A superscript T indicates that a party ties for winning a district.

• 		Red	Orange	Green	Blue
  		Party	Party	Party	Party
  	District	Voters	Voters	Voters	Voters
  	Brown
  	4	5W	4	3
  	Red	4	5T	2	5T
  	Orange	4T	4T	4T	4T
  	Yellow	3	4	3	6W
  	Green	3	3	4W	2
  	Blue	3	2	5	6W
  	Purple	6W	5	5	4
  	Pink	3	5T	5T	3
  	Gray	6W	3	4	3

• The table below shows the total number of districts that each party wins outright (W) and ties for winning (T). It also shows the total points earned by each party assuming that 3 points are earned when a party wins a district outright and 1 point is earned when a party ties for winning a district. In the final tally, the Blue Party wins this game with 8 points (2 outright wins+2 ties).

• 		# Outright	# (T)
  	Party	Wins (W)	Ties	Points

  Red

  	2	1	7 (=3*2 + 1)
  	Orange	1	3	6 (=3*1 + 3)
  	Green	1	2	5 (=3*1 + 2)
  	Blue	2	2	8 (=3*2 + 2)

Components
• The components for this game are similar to those used in Example 3. They are as follows.
• • A region 902, which may be a square in shape and be divided into 36 sector placeholders.
  • 36 sector tiles 910 (number of sectors with quantity of each icon [Red Lightning Bolt, Orange Sun, Green Diamond, Blue Moon] is in parentheses below)

• [1, 1, 1, 1] (2)	[2, 1, 1, 0] (1)	[1, 2, 1, 0] (1)	[1, 1, 2, 0] (1)	[2, 1, 0, 1] (1)	[1, 2, 0, 1] (1)	[1, 1, 0, 2] (1)
  [4, 0, 0, 0] (1)	[2, 0, 1, 1] (1)	[1, 0, 2, 1] (1)	[1, 0, 1, 2] (1)	[0, 2, 1, 1] (1)	[0, 1, 2, 1] (1)	[0, 1, 1, 2] (1)
  [0, 4, 0, 0] (1)	[2, 2, 0, 0] (1)	[2, 0, 2, 0] (1)	[2, 0, 0, 2] (1)	[0, 2, 2, 0] (1)	[0, 2, 0, 2] (1)	[0, 0, 2, 2] (1)
  [0, 0, 4, 0] (1)	[3, 1, 0, 0] (1)	[1, 3, 0, 0] (1)	[3, 0, 1, 0] (1)	[1, 0, 3, 0] (1)	[3, 0, 0, 1] (1)	[1, 0, 0, 3] (1)
  [0, 0, 0, 4] (1)	[0, 3, 1, 0] (1)	[0, 1, 3, 0] (1)	[0, 3, 0, 1] (1)	[0, 1, 0, 3] (1)	[0, 0, 3, 1] (1)	[0, 0, 1, 3] (1)

• • 9 rectangular home base markers 906 (one for each of 9 districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)
  • 162 expansion markers 908 (18 for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray)
  • 36 scoring tokens 606 (nine in each of the colors red, orange, green, and blue)
  • 2 ten-sided dice 618 (1 black, 1 white), each showing the values 0-9
  • Scoreboard 916 (FIG. 43)
Game Setup
• The setup is similar that in Example 3. The main difference is that, in this game, four scoring tokens—one for each color—are stacked on the position “No. Voters for Each Party=0” in each district's portion of the scoreboard.
Scoreboard and Scoring Tokens
• FIG. 43 shows a scoreboard 916 that may be used in strategy game 900. Scoreboard 916 has nine rows 918, one for each district to be formed. During the game, the number of voters that support each political party in each district is indicated by the position of a scoring token that matches the party's color in the row 918 for that district. For example, consider the final position shown in FIG. 42B. In this position, the total number of (Red Lightning Bolts, Orange Suns, Green Diamonds, and Blue Moons) in the Brown District is (4, 5, 4, 3) respectively. To indicate this, a (red, orange, green, blue) scoring token should be placed at position “No. Voters for Each Party”=(4, 5, 4, 3) respectively in the Brown District row 918 of the scoreboard 916. The positions of the scoring tokens may be updated after every turn.
Playing the Game
• Play may include of the following three phases. The second phase is optional.
• • 1. Build political districts
  • 2. Run an election (optional)
  • 3. Identify the winner
Phase 1: Build Political Districts
• This phase proceeds exactly as in Example 3. The only difference is that the turns alternate among up to four players instead of two.
Phase 2: Run an Election
• Phase 2 is somewhat different in a four-party version of strategy game 900 as compared to strategy game 500 in Example 3. In this phase of the game, the political status in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice. In particular, the three-step procedure below (A-B-C) is performed for each district beginning with the Brown District.
• (A) The table below is used to convert the district's political status into a probability of each party winning an election in the district. This is done by (1) ranking the parties according to voter support in the district (i.e. deciding which party is in 1^st, 2^nd, 3^rd and 4^th place); (2) computing the difference in voter support between the parties; and (3) finding the appropriate row in the table below. If two or more parties have the same voter tally, their ranking is inconsequential but the players must still (arbitrarily) rank them. For example, if the political status of the Green district in a four-player game is (3 Red Voters, 3 Orange Voters, 4 Green Voters, 2 Blue Voters), then the Green (Red, Orange, Blue) Party is in 1^st (2^nd, 3^1d, 4^th) place, the difference between 1^st and 2^nd place is 1 voter, the difference between 2^nd and 3^rd place is 0 voters, and the difference between 3^rd and 4^th place is 1 voter. In this case, the Green (Red, Orange, Blue) Party has a 60% (20%, 20%, 0%) chance of winning an election in the Green District. In a 2-3 player game, parties not represented by an active player are not ignored. These parties can still win districts and earn points at the end of the game. However, such parties are not allowed to win the game. Only a party represented by an active player may win the game.

• 	Winning Percentage for Party in
  District Political Status	1st/2nd/3rd/4th Place
  Four parties have same number of voters	25/25/25/25
  Three parties tied with the most voters	33/33/33/0
  Two parties tied with the most voters and their	45/45/10/0
  voter count exceeds party in 3rd place by 1
  Two parties tied with the most voters and their	50/50/0/0
  voter count exceeds party in 3rd place by at least 2
  One party is alone in the lead and two parties have	60/20/20/0
  one less voter than the leader
  One party is alone in the lead and one party has	70/30/0/0
  one less voter than the leader
  One party is alone in the lead and its voter count	100/0/0/0
  exceeds party in 2nd place by at least 2

• (B) Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The value on the black (white) die is the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100. In the unlikely event that the result is 100 and three parties are tied for having the most voters in the district, the dice should be re-rolled until a result below 100 is obtained.
• (C) The winner of the district's election is then determined by comparing the random number to the parties' winning percentages in the district. If the random number is less than or equal to the winning percentage of the 1^st place party, the 1^st place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1^st and 2^nd place parties, the 2^nd place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1^st, 2^nd, and 3^rd place parties, the 3^rd place party wins the district election. If the 1^st, 2^nd, and 3^rd place parties have the same number of voters in a district and the result is 100, the dice are re-rolled until a value below 100 is obtained. Otherwise, if the random number is greater than the sum of the winning percentages of the 1^st, 2^nd and 3^rd place parties, the 4^th place party wins the district election. In the example described in step A, the Green Party wins the Green District election if the random number is any value from 1-60; the Red Party wins the Green District election if the random number is any value from 61-80; and the Orange Party wins the Green District election if the random number is any value from 81-100. All scoring tokens are then removed from that district's portion of the scoreboard, and a single marker matching the color of the party that wins the election that is placed on the “Red Wins,” “Orange Wins,” “Green Wins,” or “Blue Wins” square in that district's portion of the scoreboard.
• The above procedure is repeated for each of the nine political districts.
Illustrative Example
• We consider the (4-player) game whose final board position is shown in FIG. 42B. The scoreboard for this position is shown in the section “Game Summary” above. Note that a 1^st place party automatically wins a district election (with 100% chance) if it has at least two more voters than the 2^nd place party. Thus, in the game at hand, the Blue Party automatically wins the Yellow District, and the Red Party automatically wins the Gray District.
• Dice 618 are then rolled to determine the winners of the seven districts in which there is
• not an automatic winner. The election results are summarized in the table below. In the table, the dice rolls are shown in the “Dice Roll” column, and “R,” “O,” “G,” and “B” refer to the Red, Orange, Green, and Blue Party respectively. The overall result is that the Blue (Red, Green) Party wins the elections in three (two, four) districts.

• 	Red	Orange	Green	Blue	Party	Winning
  	Party	Party	Party	Party	Ranking	Percentage	Dice	Election
  District	Voters	Voters	Voters	Voters	(1st, 2nd, 3rd, 4th)	1st/2nd/3rd/4th	Roll	Result

  Brown

  	4	5	4	3	O, R, G, B	60/20/20/0	81	Green Wins
  Red
  	4	5	2	5	O, B, R, G	45/45/10/0	45	Orange Wins
  Orange
  	4	4	4	4	R, O, G, B	25/25/25/25	34	Orange Wins
  Yellow
  	3	4	3	6	B, O, R, G	100/0/0/0	—	Blue Wins
  Green
  	3	3	4	2	G, R, O, B	60/20/20/0	2	Green Wins
  Blue
  	3	2	5	6	B, G, R, O	70/30/0/0	100	Green Wins
  Purple
  	6	5	5	4	R, O, G, B	60/20/20/0	57	Red Wins
  Pink
  	3	5	5	3	O, G, R, B	50/50/0/0	66	Green Wins
  Gray
  	6	3	4	3	R, G, O, B	100/0/0/0	—	Red Wins

Phase 3: Identify the Winner(s)
• In the game's final phase, the overall winner is identified. If phase 2 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie. For example, the Green Party wins the game shown in the table above.
• If phase 2 was not played, the scoreboard is used to identify the party that controls each district. Two or more parties jointly control a district if they tie for having the greatest number of voters in the district. If there is no tie, the party with the greatest number of voters in the district solely controls the district. Each party receives three points for each district that it solely controls and one point for each district that it jointly controls. The player whose party has more points than any other player's party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win. A party not represented by a player may not win the game. Refer to section “Game Summary” above to see who wins the game whose final position is shown in FIG. 42B if phase 2 is not played.
Example 12
• This example encompasses several games that are larger versions of the game described in Example 11. These larger versions are played with more sectors than Example 11 but are otherwise very similar to Example 11. The relationship of these games to Example 11 is analogous to the relationship of Examples 4-8 to Example 3.
• The taxonomic codes for five possible games included in this example are listed below. All games are analog, multi-player games with square sectors and a political focus in which four types of elements—namely four political parties—are present, the game proceeds according to alternating turn-based play, there are 2-4 players, and moves in categories “E,” “X,” and “R” are allowed.
• • A/G/S2/P/4/12/U/2-4/EXR
  • A/G/S3/P/4/15/U/2-4/EXR
  • A/G/S4/P/4/15/U/2-4/EXR
  • A/G/S5/P/4/21/U/2-4/EXR
  • A/G/S6/P/4/21/U/2-4/EXR
• The aforementioned five games get progressively larger with 72, 90, 121, 169, and 210 sectors respectively.
• The move types allowed in the aforementioned five games are identical to the move types described in Examples 4-8 respectively.
• The main difference between these games and Examples 4-8 is that up to four players can play these games.
• Play of any of the above games proceeds in a manner similar to Example 11. During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. In the game's final phase, the parties that control the districts are identified, and 3 (1) points are awarded to a party that has sole (joint) control of a district. The player whose party has more points than any other player's party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win.
Example 13
• Strategy games of Example 13 are a combination of the simultaneous independent play undertaken in Example 9 and the four-party environment considered in Example 11. Such games have taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. The games proceed according to simultaneous independent play, so any number of players may participate.
Components
• In one example, the game components are highly similar to the components used in Examples 9 and 11 and shown in FIGS. 36-37 and 41-43. A timer 814 may be needed to play this game, if time constraints are being used. In addition, each player should have a copy of the same game set which contains the following items:
• • 36 sector tiles 910 having the same markings as the sectors in Example 11
  • 80 expansion markers 908 (8 Brown, 10 Red, 10 Orange, 8 Yellow, 10 Green, 10 Blue, 8 Purple, 8 Pink, 8 Gray)
  • 36 scoring tokens 812 (4 for each of the nine district colors)
  • A scoreboard 916 (same as in FIG. 43)
  • A game board 818 (similar to FIG. 37 but with more scoring rows on the left side)
    Setup (about 10 minutes)
• The setup is very similar to that in Example 9, but with each player laying out a region 902 having sectors 904. Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her playing region 902 and organizes piles of markers within his/her playing area to prepare for what follows.
Playing the Game
• Play may include four phases, each having a different pre-defined goal, as listed below. Phases 2-4 are optional.
• • 1. Build political districts that maximize the Red Party's advantage
  • 2. Build political districts that maximize the Orange Party's advantage
  • 3. Build political districts that maximize the Green Party's advantage
  • 4. Build political districts that maximize the Blue Party's advantage
• Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the pre-defined goal. The main goal in each phase is to create 9 political districts—i.e. a district plan—in which the concerned party controls as many districts as possible. A party controls a district if it has strictly more voters in a district than any other party. Each player's secondary goal is to maximize the total amount—summed over the districts controlled by the concerned party—by which the concerned party leads its closest adversary in the districts that it controls.
• At the end of each phase of the game, each player tracks his/her score with respect to the pre-defined goals above by placing markers on squares in relevant rows 828 of his/her game board 818. Penalties are assessed if a player's district plan violates requirement A or B (see Example 9).
Identifying the Winner(s)
• The winner of the game is identified, by process of elimination, by looking at the markers on the rows 828 of each player's game board 818. These markers show the scores for up to eight items:
• • 1. Number of districts controlled by the Red Party in phase 1
  • 2. Total amount by which the Red Party controls its districts in phase 1
  • 3. Number of districts controlled by the Orange Party in phase 2 (if phase 2 played)
  • 4. Total amount by which the Orange Party controls its districts in phase 2 (if played)
  • 5. Number of districts controlled by the Green Party in phase 3 (if phase 3 played)
  • 6. Total amount by which the Green Party controls its districts in phase 3 (if played)
  • 7. Number of districts controlled by the Blue Party in phase 4 (if phase 4 played)
  • 8. Total amount by which the Blue Party controls its districts in phase 4 (if played)
• If all phases are played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3+5+7 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4+6+8 is eliminated. Any player not eliminated wins the game.
Example 14
• Strategy games of Example 14 combine the (optionally) time-limited, simultaneous independent play undertaken in Example 10 and the four-party environment considered in Example 11. Such games have a taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate.
Components
• In one example, the game components are nearly identical to those in Example 13. The only difference is that a few additional markers are needed to track the final score on the game board.
• Setup (about 10 Minutes)
• The setup is very similar to that in Example 9. Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her region.
Scoreboard and Scoring Tokens
• The scoreboard and scoring tokens may be the same as in Example 11.
Playing the Game
• Play proceeds just as in Example 10. As long as time has not expired, players may use their markers, scoreboard, and game board as desired to try to achieve the desired goal. Each player's main goal is to create 9 political districts—i.e. a district plan—in which all four parties control the same number of districts. A party controls a district if it has strictly more voters in a district than any other party. Each player's secondary goal is to equalize the total amount—summed over the districts controlled by each party—by which each party leads its closest adversary in the districts that it controls. Each player's tertiary goal is to maximize the number of districts in which all four parties have the same number of voters.
• At the end of play (e.g., when time expires), each player tracks his/her score with respect to the 3 goals above by placing markers on squares in the left part of his/her game board. Penalties are assessed if a player's district plan violates requirement A or B (see Example 9).
Identifying the Winner(s)
• The winner of the game is identified by looking at the markers on the left side of each player's game board. These markers show the scores for eleven items:
• • 1. Number of districts controlled by the Red Party
  • 2. Total amount by which the Red Party controls its districts
  • 3. Number of districts controlled by the Orange Party
  • 4. Total amount by which the Orange Party controls its districts
  • 5. Number of districts controlled by the Green Party
  • 6. Total amount by which the Green Party controls its districts
  • 7. Number of districts controlled by the Blue Party
  • 8. Total amount by which the Blue Party controls its districts
  • 9. The magnitude of the difference between the highest value among items 1, 3, 5, and 7 and the lowest value among items 1, 3, 5, and 7.
  • 10. The magnitude of the difference between the highest value among items 2, 4, 6, and 8 and the lowest value among items 2, 4, 6, and 8.
  • 11. The number of districts in which all parties have the same number of voters.
• The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 9 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 10 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 11 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 9), all players lose.
Example 15
• Strategy games of Example 15 have taxonomic code A/G/T1/P/2/9/U/2/EXR and are triangular-sector games similar to the square-sector games of Example 3. They are analog, multi-player games with 54 sectors in the shape of an equilateral triangle. They have two types of elements—such as two political parties—and nine districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
Game Summary
• FIGS. 44-46 illustrate one example of a strategy game 1600. In this game, two players representing opposing political parties—Red and Blue—vie for political control of a hexagonal region 1602 by competitively creating nine political districts out of 54 triangular sectors in alternating, turn-based fashion. The region 1602 may be divided into 54 sector placeholders 1608, and sector tiles 1606 may be laid out, on each sector placeholder 1608, to form sectors 1604 as part of the game set-up. Each sector 1604 may represent a community, and the set of elements on the sector tile 1606 for each sector 1604 includes a type (e.g., favored political party) and quantity (e.g., voter margin by which the indicated party is favored). The strategy game 1600 may also include a set of markers 1610, divided into marker subsets 1612 each of which represents a district. Each marker subset may include a home base marker 1614 and at least one expansion marker 1616. The number of marker subsets 1612 preferably equals the number of districts to be formed during the game. The total number of home base markers 1614 and expansion markers 1616 for each marker subset 1612 should be sufficient to form districts of appropriate size for the game. The strategy game 1600 may further include a scoreboard 1618 (same as FIG. 13), which may have one row 1624 for each district to be formed, and one scoring token 1620 for each district to be formed. The strategy game 1600 may also include two dice 1622, which may be ten sided dice, each having a different color such as one black and one white.
• During the first phase of strategy game 1600, players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of strategy game 1600, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice 1622. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game.
• FIG. 45 shows one possible initial position for a strategy game 1600, in which the sector tiles 1606 have been placed on the region, with one on each sector placeholder 1608, to form sectors 1604. FIG. 46 shows one possible final position for a game having the initial position shown in FIG. 45, with the home base markers 1614 and expansion markers 1616 placed on the region 1602, one per sector 1604, to form nine districts (indicated by bold lines). The table below shows the final result for the final position shown in FIG. 46. If phase 2 is not played, the Red Party wins the game by a score of 5 districts to 3 districts (with one tied district):

• 		Party in	By How
  	District	Control	Much

  Brown

  Blue

  	12
  	Red	Red		8
  	Orange	Neither		0
  	Yellow	Red		2
  	Green	Red		1
  	Blue	Blue		11
  	Purple	Blue		8
  	Pink	Red		6
  	Gray	Red		14

Game Components
• The game components are as follows:
• • 54 triangular sector tiles 1606 (quantity of each sector tile type is shown in parentheses below):

• +9 Red (3)	+5 Red (3)	+1 Red	(2)	+3 Blue (3)	+7 Blue (3)
  +8 Red (3)	+4 Red (3)	0	(2)	+4 Blue (3)	+8 Blue (3)
  +7 Red (3)	+3 Red (3)	+1 Blue	(2)	+5 Blue (3)	+9 Blue (3)
  +6 Red (3)	+2 Red (3)	+2 Blue	(3)	+6 Blue (3)

• • 9 home base markers 1614 (one for each of 9 districts: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray)
  • 180 expansion markers 1616 (20 for each of the nine districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray)
  • 9 scoring tokens 1620 (one for each of 9 districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray), each being cube-shaped with faces showing the numbers 10, 20, 30, 40, and 50
  • 2 ten-sided dice 1622 (1 black, 1 white), each showing the values 0-9
  • A scoreboard 1618
Playing the Game
• The game has three phases and plays in a manner similar to Example 3.
• In phase 1, players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. Forfeiting a turn (i.e. passing on a turn) is not allowed. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 60 moves—30 by each player—are made in phase 1.
• Phase 1 is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 9 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Phase 1 concludes when no legal moves exist.
• The rules may allow districts to be formed by the following six types of moves. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
• • 1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (a′) The sector must be at least three steps away from all previously placed home base markers. (b′) There must be space to grow this district to size 6 or more. Sectors 1 and 7 in FIG. 44 are five steps away from each other.
  • 1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (c′) The sector must be in the largest open space on the board. An open space is a set of connected vacant sectors. (d′) Among the sectors satisfying this criterion, the sector must be the farthest, in number of steps, from a previously placed home base marker.
  • 2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e′) The district must remain connected. (f) The district's new size—including this new sector and any sectors that are captured—must not be greater than 6 sectors. (g′) No district may be trapped.
  • 2A Expand a district by placing one of its expansion markers on a vacant sector. This move must meet three requirements. (e′) The district must remain connected. (h′) Only the smallest expandable district may be expanded. (i′) No sectors may be captured.
  • 3 (Same as in Examples 3-8 and 11-12)
  • 3A (Same as in Examples 3-8 and 11-12)
• Phases 2-3 are played almost exactly as in Example 3.
Rules for a Symmetric Game
• A symmetric version of this game may be played if players are concerned about bias in the initial sector arrangement. The purpose of a symmetric game is to remove bias from the initial sector arrangement and thereby give each party—Red and Blue—a fair chance of winning the game.
• A symmetric game has three additional rules compared to a non-symmetric game. Rule 1 guarantees a symmetrical initial sector arrangement, whereas rules 2 and 3 minimize the possibility of a symmetric board position during play. The three rules are as follows.
• • 1. During the game setup, the sector tile arrangement must be counter-symmetric with respect to the dot in the center of the region. A counter-symmetric sector arrangement is one in which the two sectors comprising every pair of diametrically-opposed sectors—i.e. every two sectors that are on exact opposite sides of the region—have opposite voter margins—for example “+4 Red” and “+4 Blue.” The procedure may be nearly identical to that described in the section “Rules for a symmetric game” in Example 3.
  • 2. During the second and third moves of phase 1 (i.e. the 1^st move made by the player who goes second, and the 2^nd move made by the player who goes first), no marker may be placed on sector that is diametrically opposed to a sector on which a marker has already been placed.
  • 3. During phase 1, any move that would result in a district that (a) has size six and (b) coincides with the hexagon in the center of the region is strictly forbidden.
Example 16
• Strategy games of Example 16 are larger versions of the games described in Example 15. These games have taxonomic code A/G/T2/P/2/12/U/2/EXR. They are analog, multi-player games with 96 triangular sectors in which 12 districts are formed. Such games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• In one example, a strategy game may be very similar to Example 15. In such a game, two players—Red and Blue—vie for political control of the hexagonal region by competitively creating 12 political districts (whose average size is 8) out of 96 triangular sectors. The sectors may be pre-established on the region, or formed by placing one sector tile on each of 96 sector placeholders on the region. The game can be played with any 96 triangular sectors, though it is preferred that the sets of sectors favoring red and blue be identical—for example 6 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (96 sectors total). Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves-50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• The six types of legal moves are summarized below.
• • 1 Establish a district on a sector that is at least 3 steps away from all other home base markers. There must be space to grow this district to size 8 or more.
  • 1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g′) no district is trapped.
  • 2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 21 or less.
  • 3 (Same as in Examples 3-8, 11-12, and 15)
  • 3A (Same as in Examples 3-8, 11-12, and 15)
Example 17
• Strategy games of Example 17 are even larger versions of the games described in Example 15. These games have taxonomic code A/G/T3/P/2/15/U/2/EXR. They are analog, multi-player games with 150 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games may proceed according to alternating, turn-based play; may have two players; and moves in categories “E,” “X,” and “R” may be allowed.
Game Summary
• FIGS. 47-48 illustrate elements of a strategy game 1000. In this game, two players representing opposing political parties—Red and Blue—vie for political control of a hexagonal region 1002 by competitively creating fifteen political districts out of 150 triangular sectors 1004 in alternating, turn-based fashion. The region 1002 may be divided into 150 sector placeholders, and sector tiles 1006 may be laid out, on each sector placeholder, to form sectors 1004 as part of the game set-up. Alternatively, each sector 1004 may be pre-established on the region 1002. Each sector 1004 may represent a community, and the set of elements 1008 in each sector 1004 includes a type (e.g., favored political party) and quantity (e.g., voter margin by which the indicated party if favored). The strategy game 1000 may also include a set of markers, divided into marker subsets that each represents a district. Each marker subset may include a home base marker 1010 and at least one expansion marker 1012. The number of marker subsets preferably equals the number of districts to be formed during the game. The total number of home base markers 1010 and expansion markers 1012 for each marker subset should be sufficient to form districts of appropriate size for the game. The strategy game 1000 may further include a scoreboard 1014, which may have one row 1016 for each district to be formed, and scoring tokens sufficient to track the score during play. The strategy game 1000 may also include two dice (which may be identical to dice 1622).
• During the first phase of the game, players build the political districts by assigning sectors 1004 to political districts, one sector per turn. They may also reassign sectors during a turn from large districts to small districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip the game's second phase.
Game Components
• The game components may be as follows:
• • 150 triangular sector tiles 1006 (quantity of each sector type is shown in parentheses below):

• +9 Red (8)	+5 Red (8)	+1 Red	(8)	+3 Blue (8)	+7 Blue (8)
  +8 Red (8)	+4 Red (8)	0	(6)	+4 Blue (8)	+8 Blue (8)
  +7 Red (8)	+3 Red (8)	+1 Blue	(8)	+5 Blue (8)	+9 Blue (8)
  +6 Red (8)	+2 Red (8)	+2 Blue	(8)	+6 Blue (8)

• • 15 home base markers 1010 (one for each of the 15 districts—Brown, Red, Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray, Black, White, Gold)
  • 450 expansion markers 1012 (30 for each of the aforementioned 15 districts)
  • 60 scoring tokens (four for each of the aforementioned 15 districts)
  • 2 ten-sided dice (1 black, 1 white), each showing the values 0-9
  • Scoreboard 1014 (FIG. 48)
Playing the Game
• The play is very similar to Example 15 but is more challenging owing to the many sectors. Play may include the following three phases. The second phase is optional.
• • 1. Build political districts
  • 2. Run an election (optional)
  • 3. Identify the winner
Phase 1: Build Political Districts Summary
• During the first phase of strategy game 1000, players take turns assigning/reassigning sectors to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector 1004 to a political district is accomplished by placing a home base marker 1010 or expansion marker 1012 on a sector tile. Reassignment of a sector 1004 from a district to another district may be accomplished by changing the color of the marker on a sector tile. At the end of the first phase, 15 political districts (e.g., Brown, Red, Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray, Black, White, and Gold) will be formed on the region 1002.
• Each district evolves in the same general way. Initially, it is formless. At some point, it is established when its home base marker 1010 is placed on a vacant sector 1004. It is then expanded whenever one of its expansion markers 1012 is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, it may be resized so its size is more similar to neighboring districts by reassignment of sectors 1004.
• The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There may not be any district size requirement. However, the rules may encourage or require the creation of districts of size 10 (meaning that each district is formed from 10 sectors 1004).
• Importantly, all marker subsets and all sectors are available to all players. No player “owns” any marker subset or sector. As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector from that district to another district.
• FIG. 47 illustrates one possible final position at the conclusion of game play, with fifteen districts formed by the home base markers 1010 and expansion markers 1012 (the boundaries of which are indicated by bold lines).
Details
• Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard.
• All moves must be of type 1, 1A, 2, 2A, 3, or 3A. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing (i.e. already established) district. Moves of type 3 and 3A resize two adjacent districts. “A” means “alternate move.”
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Phase 1 concludes when no legal moves exist.
• The six types of legal moves are as follows. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
• • 1 Establish a district on a sector that is at least 4 steps away from all other home base markers. There must be space to grow this district to size 10 or more connected sectors.
  • 1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board).
  • 2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 10 or less, and (g′) no district is trapped.
  • 2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 31 or less.
  • 3 Reassign a sector from one district to another. (Same as Examples 3-8, 11-12, 15-16)
  • 3A Reassign a sector from one district to another. (Same as Examples 3-8, 11-12, 15-16)
Phase 2: Run an Election
• The second phase of strategy game 1000 is optional. In the second phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the dice.
• Each district shown in the final position (e.g., in FIG. 47) is individually considered. First, using the table below, the political margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district.

• 	Political	Winning
  	Margin	Likelihood

  	0	50%
  	+1	59%
  	+2	67%
  	+3	74%
  	+4	80%
  	+5	85%
  	+6	89%
  	+7	92%
  	+8	94%
  	+9	96%
  	+10	98%
  	+11	99%
  	+12 or more	100%

• Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. See the section “Phase 2: Run an election” in Example 3 for more details.
• The random number is then compared to the winning percentage. If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. If both parties have a 50% chance of winning, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district's row in the scoreboard.
• The above procedure is repeated for each of the 15 political districts.
Phase 3: Identify the Winner
• In the game's final phase, the winner is identified. If phase 2 is played, the winner is the player whose party wins eight or more district elections.
• If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie.
Rules for a Symmetric Game
• As with many previous examples, strategy game 1000 can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game.
• A symmetric game for strategy game 1000 has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 reduce the possibility of a symmetric position during play. The three rules are as follows.
• • 1. During the game setup, the sector arrangement must be counter-symmetric with respect to an imaginary dot in the center of the region. The procedure for doing this may be nearly identical to that described with respect to Example 3.
  • 2. During the 2^nd, 3^rd, 4^th, and 5^th moves of phase 1 (i.e. the 1^st and 2^nd moves made by the player who goes second, and the 2^nd and 3^rd moves made by the player who goes first), no marker may be placed on a sector that is diametrically opposed to a sector on which a marker has already been placed.
  • 3. During phase 1, no move may result in a district that (a) has size ten, (b) has a political margin of zero, and (c) entirely covers the small hexagon at the center of the region 1002.
Example 18
• Strategy games of this Example 18 are even larger versions of the games with hexagonal regions and triangular sectors described in Examples 15-17. These games have taxonomic code A/G/T4/P/2/27/U/2/EXR. They are analog, multi-player games with 216 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 27 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• In one example, two players—Red and Blue—vie for political control of giant hexagonal region by competitively creating 27 political districts (whose average size is 8) out of 216 triangular communities. The game can be played with any 216 triangular sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (192 sector tiles); 8 each of sector tiles “+1 Red” and “+1 Blue” (16 sector tiles); and 8 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
• Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 27 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 27^th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
• This game has the same sector shape and same final average district size as Example 16. Thus, the details for the six types of legal moves—listed below—are identical to Example 16.
• • 1 Establish a district on a sector that is at least 3 steps away from all other home base markers. There must be space to grow this district to size 8. (Same as Example 16)
  • 1A Establish a district on a sector that ties for being the farthest, in number of steps, from a home base marker (among the sectors in the largest open space on the board). (Same as Example 16)
  • 2 Expand a district so (e′) it remains connected, (f) its new size (including any captured sectors) is 8 or less, and (g′) no district is trapped. (Same as Example 16)
  • 2A Expand the smallest expandable district so (e′) it remains connected, (i′) no sectors are captured, and (o′) its new size is 21 or less. (Same as Example 16)
  • 3 (Same as Examples 3-8, 11-12, 15-17)
  • 3A (Same as Examples 3-8, 11-12, 15-17)
Example 19
• Strategy games of Example 19 combine the simultaneous independent play undertaken in Example 9 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements (e.g., two political parties) are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate.
Components
• The game components are highly similar to the components used in Example 15, shown in FIGS. 44-46, with the addition of a game board 1100 as shown in FIG. 49, and a timer (such as timer 814 of Example 8) if time constraints will be used. Accordingly each player should have a copy of the same game set which contains the following items:
• • Region 1602
  • 54 sector tiles 1606 having the same markings as the sectors in Example 15
  • 112 expansion markers 1616 (12 Brown, 14 Red, 12 Orange, 12 Yellow, 12 Green, 14 Blue, 12 Purple, 12 Pink, 12 Gray). These are simply referred to as “markers.”
  • 9 scoring tokens (same as in Example 15)
  • Scoreboard 1618 (same as FIG. 13)
  • Game board 1100, which may have a duplicate region 1102 divided into 54 duplicate sector placeholders 1106. The game board 1100 may also have a plurality of rows 1104 configured to allow a player to track aspects of districts formed on the game board 1100.
Playing the Game
• Play may include of the following two phases. Phase 2 is optional.
• • 1. Build political districts that maximize the Red Party's advantage
  • 2. Build political districts that maximize the Blue Party's advantage
• Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the desired goal. The main goal in each phase is to create 9 political districts of equal size—i.e. a district plan—in which the concerned party controls as many districts as possible. Each player's secondary goal is to make the “voter margin in the district that the concerned party controls by the least amount” as high as possible. In the district plan, (A) each sector 1604 must be assigned to exactly one district and (B) each district must consist of six connected tiles as in FIG. 7.
• At the end of each phase of the game, each player tracks his/her score with respect to the two goals above by placing markers on the relevant square of the relevant row 1104 of his/her game board 1100. Penalties are assessed if a player's district plan violates requirement A or B above.
Identifying the Winner(s)
• The winner of the game is identified, by process of elimination, by looking at the markers on the left side of each player's game board. These markers show the scores for up to four items:
• • 1. Number of districts controlled by the Red Party in phase 1
  • 2. Lowest voter margin in the districts controlled by the Red Party in phase 1
  • 3. Number of districts controlled by the Blue Party in phase 2 (if phase 2 played)
  • 4. Lowest voter margin in the districts controlled by the Blue Party in phase 2 (if played)
• If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game.
• If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game.
Example 20
• Strategy games of Example 20 combine the simultaneous independent play undertaken in Example 10 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements—two political parties—are present and 9 districts are formed. The playing paradigm is simultaneous independent play, so any number of players may play.
Game Summary
• In one example, the game is set up exactly as described in Example 19, but the pre-defined goal of each player is to create the most balanced set of political districts. The region has an American-style, two-party political system in which one person is elected to represent each political district. At the outset, the districts are formless and the players know the political status of each sector (i.e. which party its citizens favor and by how much). During the game, players simultaneously and independently work on identical copies of the region map to create political districts that equalize the political advantage of the two parties, Red and Blue. The winner is the player who creates the most balanced set of political districts.
• The game components, setup, sectors, scoreboard, and scoring methods are identical to Example 19 except that each player uses an additional 3 gray markers to score three additional items on his/her game board.
Playing the Game
• Play proceeds as in Example 19, except with the pre-defined goal to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player's district plan must satisfy requirements A-B as stated in the section “Playing the game” in the description of Example 19.
• At the conclusion of play, the players (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player's final district plan:
• • 1. The number of districts controlled by the Red Party.
  • 2. The lowest voter margin in the districts controlled by the Red Party.
  • 3. The number of districts controlled by the Blue Party.
  • 4. The lowest voter margin in the districts controlled by the Blue Party.
  • 5. The magnitude of the difference between items 1 and 3 above (the “Red-Blue Control Differential”).
  • 6. The magnitude of the difference between items 2 and 4 above (the “Lowest Voter Margin Differential”).
  • 7. The number of tied districts (with a voter margin of 0).
Identifying the Winner(s)
• The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 19) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 19), all players lose.
Example 21
• Strategy games of Example 21 combine the alternating, turn-based play for more than two players from Example 11 with the triangular sectors of Example 15. The taxonomic code for these games is A/G/T1/P/3/9/U/2-3/EXR. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The games proceed according to alternating, turn-based play; there are 2-3 players; and moves in categories “E,” “X,” and “R” are allowed.
Game Summary
• FIGS. 50-53 illustrate elements of one example of this kind of game, a strategy game 1200. In this game, 2-3 players representing opposing political parties—Red, Green, and Blue—vie for political control of a hexagonal region 1202 by competitively creating nine political districts out of 54 triangular sectors 1204 in turn-based fashion (FIG. 50). The sectors 1204 may be pre-established on the region 1202, or they may be established during game set-up by placing sector tiles 1206 on sector placeholders (not shown) on the region 1202.
• FIG. 51 shows two examples of sector tiles 1208 and 1210 for this game. The first sector tile 1208 has a first set of elements 1212, and the second sector tile 1214 has a second set of elements 1214. As shown in FIG. 51, each sector tile 1208 and 110 contains three identical representations of the set of elements in that tile. Alternatively, each set of elements may be represented once, as shown on sector tiles 1206, or any suitable number of times in any arrangement suitable to be viewed by the players. Each set of elements includes one symbol for each element type (e.g., Red, Green, and Blue Parties) and one number showing the quantity (e.g., how many people in the sector support the Red, Green, and Blue Parties respectively). There are 6 people in each sector. The distribution of element sets for one example of a set of sector tiles is provided below, with the number of [Blue, Red, Green] Party supporters in a sector indicated in square brackets and the number of sector tiles of that type shown in parentheses:

• [6, 0, 0] (2)	[5, 0, 1] (2)	[0, 1, 5] (2)	[2, 0, 4] (2)	[3, 0, 3] (2)	[1, 1, 4] (2)	[2, 1, 3] (2)
  [0, 6, 0] (2)	[1, 5, 0] (2)	[4, 2, 0] (2)	[0, 4, 2] (2)	[0, 3, 3] (2)	[3, 2, 1] (2)	[1, 3, 2] (2)
  [0, 0, 6] (2)	[1, 0, 5] (2)	[4, 0, 2] (2)	[0, 2, 4] (2)	[4, 1, 1] (2)	[3, 1, 2] (2)	[1, 2, 3] (2)
  [5, 1, 0] (2)	[0, 5, 1] (2)	[2, 4, 0] (2)	[3, 3, 0] (2)	[1, 4, 1] (2)	[2, 3, 1] (2)

• The mechanics of this game are slightly different than in previous examples. In this game, the same types of moves—1, 1A, 2, 2A, 3, and 3A—are allowed during play, but there are additional restrictions regarding their timing. In particular, all moves of type 1, 1A, 2, and 2A must be completed during phase 1 of this game—i.e. the board must be completely full—before the first move of type 3 or 3A is allowed to be made in phase 2. The district size threshold in moves of type 2 also differs from earlier examples.
• FIG. 50 shows a possible initial board position of strategy game 1200.
• FIG. 52 shows a scoreboard 1216 that may be used in strategy game 1200. The scoring method in this game is nearly identical to that in Example 11. Each row 1218 of the scoreboard is used for tracking the political status of a different district.
• Play may include the following four phases. Phases 2-3 are optional.
• • 1. Build political districts using moves of type 1, 1A, 2, and 2A
  • 2. Rebalance district sizes using moves of type 3A
  • 3. Run an election (optional)
  • 4. Identify the winner
Phase 1: Build Political Districts Using Moves of Type 1, 1A, 2, and 2A
• This phase proceeds almost exactly like phase 1 in Example 15. The main differences are that in this game (a) turns alternate among up to three players instead of two players and (b) only moves of type 1, 1A, 2, and 2A are available.
• All moves made in phase 1 must be of type 1, 1A, 2, or 2A below.
• Play during the first phase is divided into four stages. During stage 1, only moves of type 1 and 2 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than nine districts have been established. During stage 2, only moves of type 1A and 2 are allowed. Play enters stage 3 immediately after the 9^th district is established. During stage 3, only moves of type 2 are allowed. Play enters stage 4 if no more moves of type 2 exist. During stage 4, only moves of type 2A are allowed. This phase of the game ends when every sector has been assigned to a political district.
• The four types of legal moves during this phase of the game are as follows. Note that the “5-sector district size restriction” and “no tile capturing restriction” in moves of type 2 is slightly different than in Example 15 and other previous examples.
• • 1 Establish a new district (by placing its home base marker) on a sector with the restriction that (i) the sector is at least three steps away from each previously placed home base marker and (ii) there is space to grow this district to a size of six connected sectors.
  • 1A Establish a new district (by placing its home base marker) on a sector that is within the largest connected open space on the board. An open space is an area where no sector has been assigned to a district. Among the sectors satisfying the above criterion, select a sector that ties for being the most steps away from a previously placed home base marker.
  • 2 Expand a district (by placing one of its expansion markers on a sector) so that (i) no sectors are captured, (ii) no district is trapped, (iii) the district's new size is no greater than 5 sectors, and (iv) the district remains connected.
  • 2A Expand the smallest expandable district (by placing one of its expansion markers on a sector) with the restriction that (i) no sectors are captured and (ii) the district remains connected.
Phase 2: Rebalance District Sizes Using Moves of Type 3A
• This optional phase of the game is motivated by the need to keep the populations of real-world political districts nearly equal. In this phase of the game, players take turns modifying the sector-to-district assignments in order to better equalize the district sizes (which are a proxy for the district populations).
• Players take turns beginning with the player to the left of the player who took the final turn during phase 1. During a player's turn, he/she makes a move by changing the district to which one sector is assigned. This is done by removing the (home base or expansion) marker that occupies one sector tile and replacing it with an expansion marker of a different color. The net result is that one district loses a sector and one district gains a sector. The other seven districts remain unchanged. The player then updates the scoreboard to reflect the move that has been made.
• Every move made during this phase of the game must be of type 3A:
• • 3A Reassign a community from one district (say District X) to another (say District Y). This move must meet four requirements. (j) District Y must exist prior to this move. (k) District Y must not be expandable prior to this move. (1) District X must be at least 2 sectors larger than District Y prior to this move. (m) Districts X and Y must each remain connected.
• This phase of the game concludes when no more moves of type 3A exist.
Phase 3: Run an Election
• Phase 3 in this game is optional and is very similar to phase 2 in Example 11. During this phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice.
Phase 4: Identify the Winner(s)
• In the game's final phase, the overall winner is identified. If phase 3 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie.
• If phase 3 was not played, the winner is determined by identifying the party that controls each district, i.e. the party with the most voters in each district. Each party receives 6 points for each district that it solely controls; 3 points for each district that it jointly controls with one other party; and 2 points for each district that it jointly controls with two other parties. The winner is the player whose party has more points than any other player's party. If parties represented by two or more players tie for having the most points, those players jointly win.
Symmetric Games
• A symmetric version of this game may be played if there is a desire to eliminate bias in the initial sector arrangement. A symmetric game has three additional rules compared to a regular game. Rule 1 guarantees an initial sector arrangement that is symmetric, whereas rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
• • 1. During the game setup, the sector tile arrangement must be rotationally symmetric. In a rotationally symmetric sector arrangement the number of (Blue, Red, Green) Party supporters in every sector is the same as the number of (Red, Green, Blue) Party supporters in the sector that is a 120 degree clockwise rotation from it. Such an arrangement guarantees that the initial map is unbiased, favoring no party.
    • FIG. 53 shows a rotationally symmetric sector arrangement. Bold lines distinguish three diamond-shaped portions of the board: the upper-right diamond 1220, lower-right diamond 1222, and the left diamond 1224. Note that the number of (Blue, Red, Green) Party supporters in every sector is the same as the number of (Red, Green, Blue) Party supporters in the sector that is a 120 degree clockwise rotation from it.
  • 2. The second, third, and fourth moves made during phase 1 (i.e. the 1^st move made by the player who goes second, the 1^st move made by the player who goes third, and the 2^nd move made by the player who goes first) may not involve the placement of a marker on a sector that is either a 120 degree clockwise or 120 degree counterclockwise rotation from a sector on which a marker has been placed.
  • 3. During phase 1, no move may expand an existing district so that it has size six and it coincides with the small hexagon at the center of the region.
Example 22
• Strategy games of this example encompass several games that are played with more sectors than Example 21 but are otherwise very similar to Example 21.
• The taxonomic codes for three possible games included in this example are listed below. All games are analog, multi-player games with triangular sectors and a political focus in which three types of elements—namely three political parties—are present, the game proceeds according to alternating turn-based play, there are 2-3 players, and moves in categories “E,” “X,” and “R” are allowed.
• • A/G/T2/P/3/12/U/2-3/EXR
  • A/G/T3/P/3/15/U/2-3/EXR
  • A/G/T4/P/3/27/U/2-3/EXR
• The three games above get progressively larger with 96, 150, and 216 sectors respectively.
Example 23
• Strategy games of Example 23 combine the simultaneous independent play of Example 19 with the three-party environment in Example 21. These games have taxonomic code A/G/T1/P/3/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate.
Components
• In one example, the game components are similar to the components used in Example 21. A digital or mechanical timer is needed to play this game. In addition, each player should have a copy of the same game set which contains (a) 54 sectors having the same markings as in Example 21, (b) dozens of markers, (c) a scoreboard, and (d) a game board.
Playing the Game
• Play may include the following four phases. Phases 2-4 are optional.
• • 1. Build political districts that maximize the Red Party's advantage (time limit 10 min)
  • 2. Build political districts that maximize the Green Party's advantage (time limit 10 min)
  • 3. Build political districts that maximize the Blue Party's advantage (time limit 10 min)
  • 4. Build political districts that equalize the advantage of all parties (time limit 10 min)
• Each phase 1-3 proceeds like a phase in Example 19. Phase 4 proceeds as in Example 20.
• At the end of each phase of the game, each player tracks his/her score with respect to the goal at hand by placing markers on the appropriate squares on his/her game board.
Identifying the Winner(s)
• The winner is the player who does the overall best job of achieving the goals that were pursued during the different phases of the game.
Example 24
• Strategy games of Example 24 are played with hexagonal sectors. Some examples of these games have taxonomic code A/G/H1/P/2/9/U/2/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. Taxonomic codes for five additional games included in this example are listed below. All five games are analog, multi-player games with hexagonal sectors and a political focus in which two types of elements—two political parties—are present, the game proceeds according to alternating turn-based play, and there are 2 players.
• • A/G/H2/P/2/15/U/2/EXR
  • A/G/H3/P/2/15/U/2/EXR
  • A/G/H4/P/2/21/U/2/EXR
  • A/G/H5/P/2/21/U/2/EXR
  • A/G/H6/P/2/27/U/2/EXR
• The five games above get progressively larger with 61, 91, 127, 169, and 217 sectors respectively. The shape of the region in each game is essentially a regular hexagon.
• The strategy games of this example proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
• FIG. 54 shows a region 1302 having 37 sector placeholders 1304. Sector tiles 1306, two examples of which shown in FIG. 55, may be placed on the region, one on each sector placeholder 1304, to form the sectors. Each sector has a set of elements 1308. During play, nine districts (whose average size is roughly 4) are formed from the sectors. It should be understood that regions of these games can be any size, and contain any number of sectors. The number of districts to be formed may vary depending upon the number of sectors in the region.
• Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples.
Example 25
• Strategy games of Example 25 have taxonomic code A/G/H1/P/2/9/I. They are analog, multi-player game with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate.
• Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples with two parties and simultaneous independent play.
Example 26
• Strategy games of Example 26 have taxonomic code A/G/H1/P/6/9/U/2-6/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play; there are 2-6 players; and moves in categories “E,” “X,” and “R” are allowed.
• Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 11 and 21.
Example 27
• Strategy games of Example 27 have taxonomic code A/G/H1/P/6/9/I. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, and a variety of pre-defined goals can be pursued in each phase of these games.
• Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 13-14 and 23.
Example 28
• Strategy games of Example 28 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Wisconsin. These games have taxonomic code A/G/C82/P/3/8/U/2/EXR. They are analog, multi-player games with 82 complex sectors and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and eight districts are formed. These games proceed according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed.
Game Summary
• FIGS. 56-70 illustrate aspects of a strategy game 1400. In this game, two players representing opposing political parties (Red and Blue) vie for political control of the region 1402 (i.e., the state of Wisconsin) by competitively creating eight U.S. congressional districts out of 82 sectors 1404 that largely coincide with the counties currently existing in Wisconsin. Wisconsin has a two-party political system in which one person is elected to the U.S. House of Representatives to represent each congressional district. As shown in FIG. 56, at the outset, the districts are formless and the players know the shape, location, population, and political composition of each sector (i.e. which party its citizens favor and by how much) based on the set of elements 1408 depicted in each sector 1404. During the first phase of the game, players gradually build political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. During the (optional) second phase of the game, the political margin in each district is converted into a probability of each party winning the district, and an election is simulated by rolling dice (such as dice 1622 of FIG. 44). The winner of the game is the player whose party controls more districts than his/her opponent. If both players control an equal number of districts, the result is a tie.
Components
• The components of the game are listed below:
• • Region 1402 (Wisconsin map) having 82 sectors 1404, each sector having a set of elements 1408
  • 8 marker subsets (one for each of the eight districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray), each containing a plurality of markers 1418, which may include, for example, 8 home base markers 1422 (one for each district) and 280 expansion markers 1424 (35 for each of the eight districts)
  • 64 scoring tokens (eight for each of the eight districts above)
  • 2 ten-sided dice (one black, one white) each showing values 0-9
  • Two-part scoreboard having a first part 1410 (FIG. 57) and a second part 1412 (FIG. 58).
Setup
• The players decide (yes or no) if phase 2 of the game will be played. The players then decide who plays Red and who plays Blue, and who will take the first turn.
Sectors
• As can be seen in FIG. 56, each sector on the map contains two numbers. In accordance with the key in Table 1, the first number 1414 (in a rectangle) denotes the sector's voting population (in thousands). The second number 1416 indicates the sector's voter margin (i.e. voting tendency, political margin). In the second number 1416, a black number in a white circle indicates that the sector tends to vote for the Red Party; a white number in a black circle indicates that the sector tends to vote for the Blue Party. The number itself, in this example, is the voting margin (in thousands of votes) by which the sector supported one party over the other in the 2016 U.S. presidential election. For example, a black 4 in a white circle indicates that the sector supported the Red Party by a margin of 4000 votes in the 2016 election. A white 0 in a black circle indicates that voters in the sector were evenly divided—after rounding off to the nearest thousand voters—among the two parties in the 2016 U.S. presidential election.
• In this game, the 82 sectors 1404 are used as building blocks to form eight non-overlapping political districts which together exhaust the land area of the state. The eight districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray. Initially, the political districts are formless and no sector belongs to any district. During the course of the game, players use markers 1418 to gradually assign these 82 sectors to political districts. Each sector eventually belongs to exactly one political district.
• A close inspection of the sectors 1404 as shown in FIG. 56 will reveal that the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles by 23. In other words, there are 23,000 more Red Party supporters than Blue Party supporters in the region 1402. This value agrees with the results of the 2016 U.S. presidential election. Based on this information, we say that the overall political margin in the state is “+23 for the Red Party” or “+23 Red.” Although the Red Party has a slight advantage statewide, it is highly unlikely that the Red Party will be able to maintain an advantage within every district after the region 1402 is divided into eight districts. Also note that the sum of the numbers in the black rectangles is 2793. In other words, the state's voter population is 2,793,000. Dividing this value by 8, we see that, at the game's end, the average district's voter population will be 349.125, which is most closely represented by the value 349. However, it is unlikely that any districts will have a population of exactly 349 at the game's end; most or all districts will have a population strictly greater than or less than 349.
Scoreboard
• Scoring tokens are used to display the (voter) population and political margin of every district on scoreboard (FIGS. 57-58) at all times. At the start of the game, these scoring tokens may be placed to show that the population of each district is zero and that no party has an advantage in any district.
• The scoreboard should be updated after every player takes a turn. For example, consider a moment in the game when exactly three sectors with populations 40, 71, and 48 and voting tendencies “+7 Red,” “+8 Blue,” and “+10 Red” respectively have been assigned to the Green District. In this case, the Green District's population—159—should be indicated by three green scoring tokens placed at positions “Population x100=1,” “Population x10=5,” and “Population x1=9” in the Green District's portion of the scoreboard. The Green District's current political marging—“+9 Red”—equals the difference between the sum of the numbers in the sectors that support the Red Party (e.g. 17) and the sum of the numbers in the sectors that support the Blue Party (e.g. 8) that belong to the district. The political margin favors the Red Party if there are more Red Party than Blue Party supporters in the district; it favors the Blue Party if the opposite is true. A district's political margin is indicated by four scoring tokens that show (1) which party has the majority of voters in the district, (2) the hundreds digit of the political margin, (3) the tens digit of the political margin, and (4) the ones digit of the political margin. In the case above, four green scoring tokens should be placed at the positions “Current Leader=Red,” “Political Margin x100=0,” “Political Margin x10=0,” and “Political Margin x1=9” in the Green District's portion of the scoreboard. If a sector with population 72 and voting tendency “+36 Blue” is added to this district, the district's new population is 231 and its new political margin is “+27 Blue,” so the seven green scoring tokens should immediately be moved to positions “Population x100=2,” “Population x10=3,” “Population x1=1,” “Current Leader=Blue,” “Political Margin x100=0,” “Political Margin x10=2,” and “Political Margin x1=7.”
Playing the Game
• Play may include the following three phases (the second phase is optional):
• • 1. Build political districts
  • 2. Run an election (optional)
  • 3. Identify the winner
Phase 1: Build Political Districts Summary
• This is the main phase of the game. During this phase, players take turns assigning sectors 1404 to political districts, one sector at a time, until every sector 1404 belongs to a political district. The assignment of a sector to a political district is accomplished by placing a home base marker or expansion marker on a sector. Players may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. This is done by changing the color of the marker that occupies a sector. At the end of this phase, there will be eight non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray—that together cover the state. Also, each district will be connected.
• Each district evolves in the same general way. Initially, it is formless. At some point, the district is established when its home base marker is placed on a vacant sector. (A vacant sector is a sector with no marker on it.) The district is then expanded whenever one of its expansion markers is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, the district may be adjusted so its population is more similar to neighboring districts.
• The overall process of building the political districts is relatively unrestricted. In general, any player may contribute to building any district during any of his/her turns. There is no requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a given district may be temporarily halted while players take turns establishing, expanding, and/or adjusting other districts. There is no district population requirement. However, the rules encourage the creation of districts whose population is close to the average value of 349.
Details
• Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn is not allowed.
• The rules of this game provide that all moves made during this phase of the game must be of type 1, 1A, 2, 2A, 3, or 3A below. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing district. Moves of type 3 and 3A adjust two adjacent districts by transferring a sector from one district to an adjacent district. “A” stands for “alternate move.”
• Phase 1 of the game is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than eight districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 8^th district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist.
• The six types of legal moves are as follows. Explanations for the asterisked terms are provided at the end of these descriptions.
• • 1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be at least three steps away from all previously placed home base markers. (ii) There must be space to grow this district into a connected* district with a population of at least 349. (In FIG. 59, sectors 11 and 34 are three steps away from each other. Sectors 25 and 44 are two steps away from each other. Also, sectors 57 and 72 are two steps away from each other.)
  • 1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be in the most populated open space on the board. (An open space is a connected* group of vacant sectors.) (ii) Among the sectors satisfying the criterion above, the sector must be the farthest (in number of steps) from a previously placed home base marker. Ties may be broken arbitrarily.
  • 2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (i) The district must remain connected.* (ii) The district's initial population before the move must be 348 or less. (iii) The district's new population—including the new sector and any sectors that are captured**—must be 398 or less. (iv) No district may be trapped.***
  • 2A Expand the least populous expandable district by placing one of its expansion markers on a vacant sector. The district must remain connected* and no sectors may be captured.** (A district is expandable if there is at least one vacant sector adjacent to it.)
  • 3 Reassign a sector from one district (e.g. District X) to another (e.g. District Y) by removing the District X marker from a sector and replacing it with a District Y expansion marker. This move must meet four requirements. (i) Districts X and Y must remain connected.* (ii) The populations of Districts X and Y must become strictly more balanced. In other words, before this move is made, the population of X must exceed the population of Y by more than the population of the reassigned sector. (iii) District Y must exist and be confined, i.e. it must not be expandable, prior to this move. In other words, there must be no vacant sectors adjacent to District Y prior to this move. (iv) The District X marker that is removed must be an expansion marker; it may not be a home base marker.
  • 3A Same as move type 3 but without requirement (iv) for move type 3.
• * See subsection entitled “Connectedness” below
• * See subsection entitled “Captured sectors” below
• ** See subsection entitled “Trapped districts” below
Connectedness
• Two sectors are adjacent—and connected—if and only if they share a common edge. For example, in the map shown in FIG. 59 (in which each of the 82 sectors 1404 has been given a reference number 1420 for ease of reference), sectors 1 and 2 are adjacent; sectors 35 and 45 are adjacent; and sectors 69 and 74 are adjacent. However, sectors 1 and 8 are not adjacent; sectors 44 and 56 are not adjacent; and sectors 30 and 41 are not adjacent. By analogy the territory consisting of sectors 25, 33, 44, and 54 is connected, and the territory consisting of sectors 57, 64, 68, and 72 is connected. The territory consisting of sectors 59 and 68-70 is not connected.
• In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X (i.e. a sequence of adjacent sectors that all belong to District X) connecting those two sectors.
Captured Sectors
• A set of connected, vacant sectors is captured if it is (i) surrounded by the edge of the board on one side and a single district on the other side or (ii) entirely surrounded by a single district. In FIG. 59, sectors 60 and 76 are captured by the Green District (consisting of sectors 54, 61-62, and 77); sector 18 is captured by the Blue District (consisting of sectors 17, 19, and 24); sector 41 is captured by the Gray District (consisting of sector 40); sector 29 is captured by the Yellow District (consisting of sectors 23, 28, and 30); and sectors 73-75 are captured by the Red District (consisting of sectors 69-70, 72, and 81).
• A move of type 2 which captures one or more sectors is allowed if the district's initial population before the move is 348 or less and the district's new population—including the sector where the marker is placed and any sectors that are captured—is 398 or less. All other moves that capture sectors are forbidden. For example, in FIG. 59, it is permissible to add sector 54 to a district consisting of sectors 61-62 and 77. In this case, the district immediately grows to include sectors 54, 60-62, and 76-77 after the move is made, and the district's new population is 68 which is well below the 398 threshold (FIG. 56 shows the sector populations). However, it is not permissible to add sector 72 to a district consisting of sectors 69-70 and 81 because the expanded district—which after capturing three sectors would consist of sectors 69-70, 72-75, and 81—would have a population of 569 which is well above the 398 threshold.
• Any sectors that are captured during a legal move of type 2 are immediately assigned to the district that has captured them. Expansion markers are immediately placed on these sectors.
Trapped Districts
• A district is trapped if (i) it (and the open spaces beside it) is surrounded either by the edge of the board on one side and a single district on the other side or by a single district on all sides and (ii) its population plus the populations of all open spaces beside it is 348 or less. In FIG. 59, the Orange District (consisting of sector 11) is trapped by the Brown District (consisting of sectors 8, 10, 12, 20, 22, 26-27, and 34) because the combined population of sectors 11 and 21 (=16) is 348 or less (FIG. 56 shows the sector populations).
• A move of type 2 which traps a district is forbidden. For example, if the Brown District consists of sectors 10, 12, 20, 22, 26-27, and 34 and the Orange District consists of sector 11, then an expansion of the Brown District to sector 8 is forbidden.
Phase 2: Run an Election
• In this phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the two 10-sided dice.
• Each district is considered one at a time. First, using the table below, the political margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+4 Blue” political margin in the Yellow District converts to a 66% chance for the Blue Party to win an election in the Yellow District.

• 	Political	Winning
  	Margin	Likelihood

  	0	50%
  	+1	54%
  	+2	58%
  	+3	62%
  	+4	66%
  	+5	70%
  	+6	73%
  	+7	76%
  	+8	79%
  	+9	82%
  	+10	85%
  	+11	87%
  	+12	89%
  	+13	91%
  	+14	93%
  	+15	95%
  	+16	96%
  	+17	97%
  	+18	98%
  	+19	99%
  	+20 or more	100%

• Next, a random number from 1-100 is produced by rolling the two 10-sided dice.
• The random number is then compared to the winning percentage (e.g. 66 for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Blue Party wins the Yellow District election if the random number is from 1-66, and the Red Party wins the Yellow District election if the random number is from 67-100. If both parties have a 50% chance of winning the election, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Winner=Blue” or “Winner=Red” square in the appropriate district's portion of the scoreboard.
• The above procedure is repeated for each political district.
Phase 3: Identify the Winner
• In the game's final phase, the overall winner is identified.
• If phase 2 is played, the winner is the player whose party wins five or more district elections. If each party wins four district elections, the result is a tie.
• If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie.
Example of Play
• An example of play is now provided, with reference to FIGS. 60-70, to illustrate the rules of the game. The initial position is shown in FIG. 56. During play, the population and voting tendency of each sector are visible to both players. However, in FIGS. 60-69, the reference number 1420 of each sector 1404 is shown instead of the voting tendency for ease of discussion.
• During stage 1, only moves of type 1, 2, and 3 are allowed. After 40 moves have been made—six of type 1 and 34 of type 2—assume the board position, with home base markers 1422 and expansion markers 1424, is as shown in FIG. 60. Note that six districts—Brown, Red, Orange, Yellow, Blue, and Purple—have been established. Two districts—Green and Gray—have not (yet) been established.
• There are 43 possibilities for the next move which must be of type 1, 2, or 3. No legal moves of type 3 exist because no district is confined. Regarding moves of type 1, it is possible to establish a new district in sector 73 or 74. Establishing a new district in another sector is not allowed because either (i) the sector is less than three steps away from a previously placed home base marker or (ii) there is not enough space to grow the new district into a connected district with a population of at least 349. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector 5, 13, 22, 21, 19, 56, 50, 46, 28, 29, 30, or 16; (b) expand the Red District to sector 28, 46, 51, 57, 53, 40, 38, or 37; (c) expand the Orange District to sector 19, 24, 31, 32, 42, 43, 54, 60, 76, 65, 63, or 56; (d) expand the Yellow District to sector 57, 67, 70, 72, or 59; or (e) expand the Blue District to sector 4, 9, 21, or 12. The Purple District may not be expanded during the next move because its population—352—is already at least 349.
• Several of the above moves of type 2 capture one or more sectors including the expansion of the (a) Brown District to sector 30 (which captures sector 16); (b) Red District to sector 40 (which captures sector 41); (c) Red District to sector 38 (which captures sectors 40 and 41); and (d) Orange District to sector 24, 31, 32, 42, 54, or 60. Note that an expansion of the Brown District to sector 9 is not allowed because the Blue District would be trapped (by the Brown District). Also, an expansion of the Yellow District to sector 53, 73, or 74 is not allowed because one or more sectors would be captured and added to the Yellow District, putting its population over the limit of 398.
• The next move played is the establishment of the Green District in sector 74. This results in the game position shown in FIG. 61. In this position, (i) there is no way to make a move of type 1 that satisfies its criteria and (ii) fewer than eight districts have been established. Thus, stage 2 of play begins.
• There are 52 possibilities for the next move which must be of type 1A, 2, or 3. No legal moves of type 3 exist because no district is confined. The feasible moves of type 2 include the 41 moves of type 2 mentioned above and (f) expanding the Green District to sector 70, 75, or 73.
• A move of type 1A requires that a new district be established in a sector that is within the most populous connected open space on the board. Among the sectors satisfying this criterion, a sector that ties for being the most steps away from a previously placed home base marker must be selected. In the current board position, the most populous open space has population 511 and consists of sectors 46, 50-51, 56-57, 63-65, 67, 70, and 75. Among these sectors, eight tie for being two steps away from a previously placed home base marker—46, 50-51, 56, 63-65, and 67—and none is three or more steps away from all previously placed home base markers. Thus, there are eight possible moves of type 1A: establish the Gray District in sector 46, 50-51, 56, 63-65, or 67.
• The next five moves in this example game are as follows (move type in parentheses):
• • 1. (2) Expansion of Red District to sector 40 (one sector captured)
  • 2. (2) Expansion of Blue District to sector 12
  • 3. (2) Expansion of Brown District to sector 30 (one sector captured)
  • 4. (2) Expansion of Orange District to sector 24 (seven sectors captured)
  • 5. (1A) Establishment of Gray District in sector 56
• The establishment of the 8^th district during move #5 above ushers in stage 3 of play. During stage 3, the next move must be of type 2 or 3A if at least one type 2 move exists. Otherwise the next move must be of type 2A or 3A. The new board position is shown in FIG. 62.
• There are 40 possibilities for the next move. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector 5, 13, 22, 21, 19, 50, 46, 28, 29, or 38 (but not 9); (b) expand the Red District to sector 28, 46, 51, 57, 53, 38, or 37; (c) expand the Orange District to sector 18, 65, 63, or 19; (d) expand the Yellow District to sector 57, 67, 70, 72, or 59 (but not 53 or 73); (e) expand the Green District to sector 70, 75, or 73; (f) expand the Blue District to sector 4, 9, 21, 22, 13, or 5; or (g) expand the Gray District to sector 50, 63, 64, 57, or 51. The Purple District may not be expanded during the next move because its population—352—is already at least 349. There are no legal moves of type 3A because no district is confined.
• After the next 23 moves in the game, the exemplary board position is shown in FIG. 63. The preceding 23 moves were all of type 2, and these moves involved the expansion of seven districts: Brown, Red, Orange, Yellow, Green, Blue, and Gray.
• The table below shows the current population of each district. Note that no feasible move of type 2 exists. This is because all districts already have a population of at least 349 and/or are confined. In particular, the only two districts with a population of 348 or less—Blue and Gray—are confined. Thus, the next move must be of type 2A or 3A.

• 	District	Population

  	Brown	366
  	Red	359
  	Orange	360
  	Yellow	355
  	Green	352
  	Blue	243
  	Purple	352
  	Gray	141

• A move of type 3A is possible—i.e. a sector may be reassigned from District X to District Y—if and only if (i) Districts X and Y remain connected after the reassignment, (ii) before the reassignment the population of X exceeds the population of Y by more than the population of the reassigned sector, and (iii) District Y is confined before the reassignment. Note that accurate district population information (as shown in the table above) is needed in order to make a correct assessment regarding item (ii) above.
• To search for moves of type 3A, we consider each district one at a time and ask if that district can “steal” a sector from another district.
• • The Brown District may not steal a sector from another district for two reasons: (A) it is the most populous district and (B) it is not confined.
  • The Red District is not confined, so it may not steal a sector from another district.
  • The Orange District is confined and its population is 6 less than the Brown District, so the Orange District could theoretically steal a sector with population 5 or less from the Brown District. However, the Brown District has no such sector that lies along the Orange-Brown border, so there is no legal way for the Orange District to steal a sector from another district.
  • The Yellow District is not confined, so it may not steal a sector from another district.
  • The Green District is not confined, so it may not steal a sector from another district.
  • There are five legal moves of type 3A in which the Blue District—whose population is much less than the other districts except the Gray District—steals a sector from a neighboring district. These include (a) reassigning sector 10, 14, or 23 from the Brown District to the Blue District and (b) reassigning sector 24 or 25 from the Orange District to the Blue District. Note that reassigning sector 20, 26, or 27 from the Brown District to the Blue District destroys the connectedness of the Brown district, so these moves are not allowed.
  • The Purple District's population is slightly less than that of the Brown, Red, Orange, and Yellow Districts, so the Purple District could theoretically steal a small sector from one of these districts. However, the Purple District does not share a border with the Brown, Red, or Yellow District. In addition, the Orange District does not have a small sector that lies along the Purple-Orange border. Thus, there is no legal way for the Purple District to steal a sector from another district.
  • There are eight legal moves of type 3A in which the Gray District—the least populous district—steals a sector from another district. These include (a) reassigning sector 51 from the Brown District to the Gray District; (b) reassigning sector 55, 62, or 65 from the Orange District to the Gray District; (c) reassigning sector 66 from the Purple District to the Gray District; (d) reassigning sector 64 from the Green District to the Gray District; (e) reassigning sector 58 from the Yellow District to the Gray District; and (f) reassigning sector 52 from the Red District to the Gray District. The following moves are not allowed because they destroy the connectedness of a district: (g) reassigning sector 45 or 50 from the Brown District to the Gray District; (h) reassigning sector 67 from the Green District to the Gray District; and (i) reassigning sector 68 from the Yellow District to the Gray District.
• To search for moves of type 2A, note that the Green District—with a population of 352—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Green District to sector 73. Overall, a total of 14 legal moves exist.
• The next five moves are as follows (move type in parentheses):
• • 1. (2A) Expansion of Green District to sector 73
  • 2. (3A) Reassignment of sector 23 from Brown District to Blue District
  • 3. (3A) Reassignment of sector 10 from Brown District to Blue District
  • 4. (3A) Reassignment of sector 24 from Orange District to Blue District
  • 5. (3A) Reassignment of sector 51 from Brown District to Gray District
• The board position is now as shown in FIG. 64. The current populations of the districts are shown in the table below. Districts with a population of 348 or less are asterisked. Note that two districts with a population of 348 or less—Brown and Gray—are not confined. (The Gray District—the least populated district—became unconfined during move #5 above. The Brown District—which was never confined—became underpopulated—with a population of 348 or less—during move #3 above.) Thus, a move of type 2 is available. The next move must therefore be of type 2 or 3A.

• 	District	Population
  	Brown	325*
  	Red	359
  	Orange	356
  	Yellow	355
  	Green	435
  	Blue	279*
  	Purple	352
  	Gray	150*

• Two moves of type 2 are available: (a) expand the Gray District to sector 46 and (b) expand the Brown District to sector 37. Six moves of type 3A are also available: (c) reassign sector 14, 29, or 20 from the Brown District to the Blue District; (d) reassign sector 25 or 31 from the Orange District to the Blue District; and (e) reassign sector 64 from the Green District to the Purple District. Note that the Brown, Red, Yellow, Green, and Gray Districts are not confined, so they may not steal a sector from another district. Also, the Purple District may not steal sector 75 from the Green District because the population imbalance would not be reduced. Overall, eight legal moves are available.
• The next five moves are as follows (move type in parentheses):
• • 1. (2) Expansion of Brown District to sector 37
  • 2. (2) Expansion of Gray District to sector 46
  • 3. (3A) Reassignment of sector 20 from Brown District to Blue District
  • 4. (3A) Reassignment of sector 31 from Orange District to Blue District
  • 5. (3A) Reassignment of sector 26 from Brown District to Blue District
• The new board position is shown in FIG. 65. The current district populations are shown in the table below. Districts with a population of 348 or less are asterisked. There are no feasible moves of type 2, so the next move must be of type 2A or 3A.

• 	District	Population
  	Brown	370
  	Red	359
  	Orange	349
  	Yellow	355
  	Green	435
  	Blue	329*
  	Purple	352
  	Gray	161*

• There are 17 possibilities for the next move. To search for moves of type 2A, note that the Yellow District—with a population of 355—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Yellow District to sector 72. Several moves of type 3A are available: (b) reassign sector 14, 29, or 34 from the Brown District to the Blue District; (c) reassign sector 32 or 33 from the Orange District to the Blue District; (d) reassign sector 64 from the Green District to the Purple District; (e) reassign sector 35 or 50 from the Brown District to the Gray District; (f) reassign sector 36 or 52 from the Red District to the Gray District; (g) reassign sector 55, 62, or 65 from the Orange District to the Gray District; (h) reassign sector 58 from the Yellow District to the Gray District; (i) reassign sector 64 from the Green District to the Gray District; and (j) reassign sector 66 from the Purple District to the Gray District.
• The next four moves are as follows (move type in parentheses):
• • 1. (3A) Reassignment of sector 64 from Green District to Gray District
  • 2. (3A) Reassignment of sector 34 from Brown District to Blue District
  • 3. (3A) Reassignment of sector 62 from Orange District to Gray District
  • 4. (2A) Expansion of Yellow District to sector 72
• The new board position is shown in FIG. 66. Note that there are no vacant sectors. Thus, all future moves will be of type 3A. Play continues until no more such moves exist. The new district populations are shown in the table below.

• 	District	Population
  	Brown	334*
  	Red	359
  	Orange	338*
  	Yellow	438
  	Green	363
  	Blue	365
  	Purple	352
  	Gray	244*

• There are 21 possibilities for the next move: (a) reassign sector 5, 13, 21, 22, or 23 from the Blue District to the Brown District; (b) reassign sector 58 or 59 from the Yellow District to the Red District; (c) reassign sector 19 or 31 from the Blue District to the Orange District; (d) reassign sector 68 or 69 from the Yellow District to the Green District; (e) reassign sector 50 from the Brown District to the Gray District; (f) reassign sector 36 or 52 from the Red District to the Gray District; (g) reassign sector 55, 61, or 65 from the Orange District to the Gray District; (h) reassign sector 58 or 68 from the Yellow District to the Gray District; (i) reassign sector 67 from the Green District to the Gray District; or (j) reassign sector 66 from the Purple District to the Gray District.
• The next two moves are as follows (move type in parentheses):
• • 1. (3A) Reassignment of sector 22 from Blue District to Brown District
  • 2. (3A) Reassignment of sector 68 from Yellow District to Gray District
• The new board position is shown in FIG. 67. Note that the new district populations—shown in the table below—are more balanced than before. Each future move will make the district populations even more balanced until, at the end of the game, no further “rebalancing” is possible. At that point, an “equilibrium” will be established, and the competition between the two players—who alternate turns and are representing the Red and Blue Parties—will end.

• 	District	Population
  	Brown	348*
  	Red	359
  	Orange	338*
  	Yellow	364
  	Green	363
  	Blue	351
  	Purple	352
  	Gray	318*

• There are five possibilities for the next move: (a) reassign sector 31 from the Blue District to the Orange District; (b) reassign sector 50 from the Brown District to the Gray District; (c) reassign sector 36 from the Red District to the Gray District; (d) reassign sector 61 from the Orange District to the Gray District; or (e) reassign sector 67 from the Green District to the Gray District.
• The next two moves are as follows (move type in parentheses):
• • 1. (3A) Reassignment of sector 36 from Red District to Gray District
  • 2. (3A) Reassignment of sector 31 from Blue District to Orange District
• The new board position is shown in FIG. 68 and the new district populations are in the table below.

• 	District	Population
  	Brown	348*
  	Red	334*
  	Orange	345*
  	Yellow	364
  	Green	363
  	Blue	344*
  	Purple	352
  	Gray	343*

• There is only one possibility for the next move: reassign sector 29 from the Brown District to the Blue District. This move is compulsory for the player who takes the next turn.
• The new board position is shown in FIG. 69 and the new district populations are in the table below. No legal moves exist in this board position, so phase 1 of play concludes. The districts that have been formed are now final. All districts are connected, but some—such as Brown, Yellow, Green, Blue, and Gray—have irregular shapes with tentacle-like protrusions. As expected, the district populations sum to 2793, and the population of the average district is 349.125=2793/8. However, no district has a population equaling 349 or 350. Three districts—Yellow, Green, and Purple—have populations exceeding 350 and five districts—Brown, Red, Orange, Blue, and Gray—have populations below 349. The district populations are not perfectly smooth, but they are relatively balanced. The population difference between the most populous district—Yellow—and least populous district—Red—is only 30. This situation is not uncommon at the end of phase 1.
• The final board position at the end of phase 1 is shown in FIG. 70. The markers played and each sector's population and voting tendency—which are always displayed to both players during the game—are shown. The population is displayed in a black rectangle and the voting tendency is displayed in a circle.
• At the end of phase 1, the scoreboard should read as shown in the box below.

• 			Political
  	District	Population	Margin

  	Brown	347	+61 Red
  	Red	334	+56 Red
  	Orange	345	+31 Blue
  	Yellow	364	+4 Blue
  	Green	363	+69 Blue
  	Blue	345	+64 Red
  	Purple	352	+30 Blue
  	Gray	343	+24 Blue
  	Overall	2793	+23 Red

• If phase 2 is not played, the game ends, and the Blue Party wins by a score of 5 districts to 3 districts.
• If phase 2 is played, an election is simulated. In the final scoreboard (see above), the political margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Gray) District translates to a winning likelihood of (100%, 100%, 100%, 66%, 100%, 100%, 100%, 100%) for the party that has the majority of voters in the district. Note that the Red Party automatically wins three districts with 100% probability and the Blue Party automatically wins four districts.
• Dice are then thrown to determine the election results for the district in which there is not an automatic winner. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in four districts. The result is a tie.

• 		Political	Winning	Dice	Election
  	District	Margin	Likelihood	Roll	Result
  	Brown	+61 Red	100%	—	Red Wins
  	Red	+56 Red	100%	—	Red Wins
  	Orange	+31 Blue	100%	—	Blue Wins
  	Yellow	+4 Blue	66%	67	Red Wins
  	Green	+69 Blue	100%	—	Blue Wins
  	Blue	+64 Red	100%	—	Red Wins
  	Purple	+30 Blue	100%	—	Blue Wins
  	Gray	+24 Blue	100%	—	Blue Wins

Example 29
• Strategy games of this Example 29 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Michigan. These games have taxonomic code A/G/C108/P/3/14/U/2/EXR. These games are analog, multi-player games with complex sectors (which may mimic actual counties of the state) and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and a pre-determined number of districts are formed. The game proceeds according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed.
Game Summary
• FIGS. 71-74 illustrate aspects of a strategy game 1500. In this game, two players representing opposing political parties (Red and Blue) vie for political control of region 1502 (e.g., Michigan) by competitively creating 14 U.S. congressional districts out of 108 sectors 1504 that largely coincide with counties that currently exist in Michigan.
• One example of an initial position is illustrated in FIG. 71. At the outset, the districts are formless and the players know the shape, location, population, and political composition of each sector 1504 (i.e. which party its citizens favor and by how much) based on the set of elements 1508 depicted in each sector 1504. During the first phase of the game, players gradually build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. During the (optional) second phase of the game, the political margin in each district is converted into a probability of each party winning the district, and an election is simulated by rolling dice (such as dice 1622 of FIG. 44). The winner of the game is the player whose party controls more districts than his/her opponent. If both players control an equal number of districts, the result is a tie. The sets of elements shown in FIG. 71 are based on the results of the 2016 U.S. presidential election, and each includes two numbers. In accordance with the key in Table 1, the first number 1514 (in a rectangle) denotes the sector's voting population (in thousands). The second number 1516 indicates the sector's voting tendency. FIGS. 72-73 show an example of a first part of a scoreboard 1510 and a second part of a scoreboard 1512 that may be included in a strategy game 1500.
Playing the Game
• The rules are very similar to Example 28. The main difference is that this game has different population thresholds for moves of type 1 and 2.
• All moves made during phase 1 must be of type 1, 1A, 2, 2A, 3, or 3A below.
• Phase 1 is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than 14 districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 14^th district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist.
• The six types of legal moves are as follows:
• • 1 Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be at least three steps away from all previously placed home base markers. (ii) There must be space to grow this district into a connected district with a population of at least 320.
  • 1A Establish a new district by placing its home base marker on a vacant sector. This move must meet two requirements. (i) The sector must be in the most populated open space on the board. (An open space is a connected group of vacant sectors.) (ii) Among the sectors satisfying the criterion above, the sector must be the farthest (in number of steps) from a previously placed home base marker. Ties may be broken arbitrarily.
  • 2 Expand a district by placing one of its expansion markers on a vacant sector. This move must meet four requirements. (i) The district must remain connected. (ii) The district's initial population before the move is 320 or less. (iii) The district's new population—including the new sector and any sectors that are captured—is 340 or less. (iv) No district may be trapped. A district is trapped if (a) it, and the open spaces beside it, is surrounded either by the edge of the board on one side and a single district on the other side or by a single district on all sides and (b) its population plus the populations of all open spaces beside it is 320 or less.
  • 2A Expand the least populous expandable district by placing one of its expansion markers on a vacant sector. The district must remain connected and no sectors may be captured. (A district is expandable if there is at least one vacant sector adjacent to it.)
  • 3 (Same as in Example 28)
  • 3A (Same as in Example 28)
Example of Play
• We now provide an example of play. After 127 moves have been made—64 by the player representing the Red Party and 63 by the player representing the Blue Party—a final position as shown in FIG. 74 may be reached, with home base markers 1518 and expansion markers 1520 placed in a manner that defines 14 districts within the region 1502. No legal moves exist in this board position, so phase 1 of play concludes. As expected, the district populations sum to 4548, and the population of the average district is 324.857=4548/14. However, only one district—Purple—has a population equaling 324 or 325.

• 			Political
  	District	Population	Margin

  	Brown	269	+65 Blue
  	Red	348	+52 Red
  	Orange
  	330	+21 Red
  	Yellow	331	+49 Red
  	Lt. Green	345	+70 Red
  	Dk. Green	328	+14 Red
  	Lt. Blue	350	+92 Blue
  	Dk. Blue	272	+24 Blue
  	Purple	324	+21 Blue
  	Pink	338	+34 Red
  	Gray	356	+2 Blue
  	Black	339	+2 Red
  	White	338	+81 Red
  	Gold	280	+108 Blue
  	Overall	4548	11

• If phase 2 is not played, the game ends, and the Red Party wins by a score of 8 districts to 6 districts.
Example 30
• Strategy games of this example 30 have taxonomic code A/G/C?/N/3/?/U/2-3/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-3 players. Each player represents a tribe. The players are provided with a playing surface illustrating a region divided into a number of sectors. The premise is that three tribes have been fighting wars against each other within the region illustrated on the board for more than a century. After significant bloodshed and no clear winner, they have decided to peacefully settle their differences by forming districts that various tribes will inhabit upon conclusion of the game.
• During the game, the players organize the region into districts such that (i) each sector is assigned in its entirety to exactly one district and (ii) each district is a single connected piece.
• At the outset the districts are formless and the players are informed of each sector's precise shape, location, and set of elements. The set of elements for each sector provides the intensity of each of three elements—rivers, plants, and mammals—in the sector.
• During play, the players take alternating turns, with each player taking a single turn before any other player takes another turn. During a player's turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two adjacent districts by returning all sectors assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 8 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist.
• The method of scoring at the end is nontrivial and relates to the suitability (i.e. habitability) of each district for each tribe.
• Tribes A, B, and C have different habitability criteria. Members of Tribe A depend on fishing for sustenance and are allergic to plants. That is, Tribe A considers rivers as a resource and plants as a hazard, and it is indifferent to mammals. Members of Tribe B depend on plants/farming for sustenance and are allergic to mammals. That is, Tribe B considers plants as a resource and mammals as a hazard, and it is indifferent to rivers. Members of Tribe C depend on hunting for sustenance and are very poor swimmers. That is, Tribe C considers mammals as a resource and rivers as a hazard, and it is indifferent to plants.
• Once the districts are finalized (i.e., when no legal moves exist), the intensity of each element in each district is computed by summing the intensities of the element in the sectors comprising the district.
• At the end of the game, each tribe receives points for each district as follows.
• • If the total intensity of rivers exceeds the total intensity of plants in a district, Tribe A considers the district habitable and receives [(river intensity)−(plant intensity)] points for that district. Otherwise, Tribe A receives zero points for that district.
  • If the total intensity of plants exceeds the total intensity of mammals in a district, Tribe B considers the district habitable and receives [(plant intensity)−(mammal intensity)] points for that district. Otherwise, Tribe B receives zero points for that district.
  • If the total intensity of mammals exceeds the total intensity of rivers in a district, Tribe C considers the district habitable and receives [(mammal intensity)−(river intensity)] points for that district. Otherwise, Tribe C receives zero points for that district.
• Each tribe's point total at the end of the game equals the sum of the points it receives in all districts. The winner is the player (i.e. tribe) with the most points at the end of the game.
Example 31
• Strategy games of this example 31 have taxonomic code A/G/C?/N/6/?/U/2-6/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-6 players. Each player represents an interplanetary transportation company. In one example, the premise is that six interplanetary transportation companies dominate the economy of the region of the Milky Way Galaxy in the year 2388. After decades of chaos in the transportation market, the companies have decided, for their mutual benefit, to set standard transportation rates within the galaxy by dividing it into districts. After the districts are formed, direct transportation between planets will only take place (i) within districts and (ii) between adjacent districts. No other direct transportation services will be offered.
• During the game, players organize the galaxy—which is already divided into C sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the entire galaxy—into a given number, D, of districts (where 2≤D<C) such that (a) each sector belongs in its entirety to exactly one district and (b) each district is comprised of a set of adjacent sectors.
• At the outset the districts are formless and the players are informed of each sector's precise shape, location, and set of elements, which in this case is a set of planets. The set of planets in a sector consists of six numbers which respectively represent the number of each planet type—agricultural, metropolitan, scholarly, industrial, medical, and ecological—in the sector.
• Players take turns in rotating fashion. During a player's turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two or three adjacent districts by returning all sectors previously assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 5 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist.
• The method of scoring is nontrivial; it relates to the profitability of each district and each pair of adjacent districts for each company.
• Each company specializes in a different kind of transportation and therefore has a different perspective on profitability. Company A specializes in transporting food and food equipment between agricultural and metropolitan planets. Company B specializes in transporting students and researchers between metropolitan and scholarly planets. Company C specializes in transporting workers and engineers between scholarly and industrial planets. Company D specializes in transporting injured workers between industrial and medical planets. Company E specializes in transporting people and medicinal plants between medical and ecological planets. Company F specializes in transporting flora and fauna between ecological and agricultural planets.
• Once the districts are finalized (i.e., once no legal moves exist), the total number of each planet type in each district is computed by summing the number of that planet type in the sectors comprising the district. Each company receives points for each district as follows.
• • Company A receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one agricultural and one metropolitan—in the district. So Company A gets [3*(no. of agricultural planets in district)*(no. of metropolitan planets in district)] points for the district.
  • Company B receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one metropolitan and one scholarly—in the district. So Company B gets [3*(no. of metropolitan planets in district)*(no. of scholarly planets in district)] points for the district.
  • Company C receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one scholarly and one industrial—in the district. So Company C gets [3*(no. of scholarly planets in district)*(no. of industrial planets in district)] points for the district.
  • Company D receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one industrial and one medical—in the district. So Company D gets [3*(no. of industrial planets in district)*(no. of medical planets in district)] points for the district.
  • Company E receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one medical and one ecological—in the district. So Company E gets [3*(no. of medical planets in district)*(no. of ecological planets in district)] points for the district.
  • Company F receives 3 points for each transportation leg that it can service in the district, i.e. 3 points for each unique pair of planets—one ecological and one agricultural—in the district. So Company F gets [3*(no. of ecological planets in district)*(no. of agricultural planets in district)] points for the district.
• Each company also receives points for each pair of adjacent districts (e.g. X and Y) as follows.
• • Company A gets 1 point for each transportation leg that it can service between Districts X and Y. So Company A gets [(no. of agricultural planets in X)*(no. of metropolitan planets in Y)]+[(no. of agricultural planets in Y)*(no. of metropolitan planets in X)] points for this pair of districts.
  • Company B gets 1 point for each transportation leg that it can service between Districts X and Y. So Company B gets [(no. of metropolitan planets in X)*(no. of scholarly planets in Y)]+[(no. of metropolitan planets in Y)*(no. of scholarly planets in X)] points for this pair of districts.
  • Company C gets 1 point for each transportation leg that it can service between Districts X and Y. So Company C gets [(no. of scholarly planets in X)*(no. of industrial planets in Y)]+[(no. of scholarly planets in Y)*(no. of industrial planets in X)] points for this pair of districts.
  • Company D gets 1 point for each transportation leg that it can service between Districts X and Y. So Company D gets [(no. of industrial planets in X)*(no. of medical planets in Y)]+[(no. of industrial planets in Y)*(no. of medical planets in X)] points for this pair of districts.
  • Company E gets 1 point for each transportation leg that it can service between Districts X and Y. So Company E gets [(no. of medical planets in X)*(no. of ecological planets in Y)]+[(no. of medical planets in Y)*(no. of ecological planets in X)] points for this pair of districts.
  • Company F gets 1 point for each transportation leg that it can service between Districts X and Y. So Company F gets [(no. of ecological planets in X)*(no. of agricultural planets in Y)]+[(no. of ecological planets in Y)*(no. of agricultural planets in X)] points for this pair of districts.
• Each company's point total at the end of the game equals the sum of the points it receives in all districts plus the sum of the points it receives in all pairs of adjacent districts. The winner is the player (i.e. company) with the most points at the end of the game.
• From the foregoing, it will be appreciated that although specific examples have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit or scope of this disclosure. For example, even though the Examples are described as being analog, they could alternatively be digital, and any component of the games could be digitally represented. It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to particularly point out and distinctly claim the claimed subject matter.

Claims (20)

What is claimed is:

1. A strategy game for at least one player comprising:

a region comprising a bounded shape having an area divided into a plurality of sectors, wherein each sector comprises a bounded shape having an area within the region that does not overlap with any other sector, and each sector contains a set of elements, each element of the set of elements having a type and quantity;

wherein, during play, each player makes one move per turn, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.

2. The strategy game of claim 1, wherein each element of the set of elements is selected from the group consisting of: resources, hazards, and scraps.

3. The strategy game of claim 1, wherein the pre-defined goal is selected from the group consisting of: maximizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; minimizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; and maximizing a number of points earned by at least one of the players, where the number of points earned by the at least one player depends on the aggregation of the elements within each district.

4. The strategy game of claim 1, wherein the strategy game further includes at least two political parties, and the set of elements in each sector represents a margin of voters within the sector that support each political party.

5. The strategy game of claim 4, wherein the pre-defined goal is selected from the group consisting of: maximizing a portion of the given number of districts controlled by one of the political parties, and equalizing a portion of the given number of districts controlled by each of the political parties.

6. The strategy game of claim 1, further comprising a set of markers, wherein the set of markers comprises a plurality of marker subsets, each marker subset representing a district;

wherein each marker subset comprises a plurality of markers, and each marker within the marker subset is configured to be placed on a sector to assign the sector as being part of the district represented by the marker subset.

7. A method of playing a strategy game for at least one player comprising:

providing a region comprising a bounded shape having an area divided into a plurality of sectors, wherein each sector comprises a bounded shape having an area within the region that does not overlap with any other sector, and each sector contains a set of elements, each element of the set of elements having a type and quantity;

making one move per turn per player, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.

8. The method of claim 7, further comprising:

providing a set of markers, wherein the set of markers comprises a plurality of marker subsets, each marker subset representing a district;

wherein each marker subset comprises a plurality of markers, and each marker within the marker subset is configured to be placed on a sector to assign the sector as being part of the district represented by the marker subset.

9. The method of claim 8, wherein the types of moves that can be made by the at least one player include assigning a sector to a district, and the assigning comprises placing a marker from one of the plurality of marker subsets onto the sector.

10. The method of claim 7, wherein the types of moves that can be made by the at least one player includes establishing a new district by assigning a first sector to a district to create an established district.

11. The method of claim 7, wherein the types of moves that can be made by the at least one player includes expanding a district by assigning an unassigned sector to a previously established district.

12. The method of claim 7, wherein the types of moves that can be made by the at least one player includes reassigning a sector from one district to another district.

13. The method of claim 7, wherein the restrictions governing how any of the districts can be formed from the plurality of sectors include a restriction that each district must be comprised of one or more sectors.

14. The method of claim 7, wherein the restrictions governing how any of the districts can be formed from the plurality of sectors include a restriction that, at the conclusion of play, each district must have a number of sectors selected from the group consisting of: a minimum number of sectors, a maximum number of sectors, or a specific pre-defined number of sectors.

15. The method of claim 7, wherein the restrictions governing how any of the districts can be formed from the plurality of sectors include a restriction that each district must be connected.

16. The method of claim 7, wherein the restrictions governing how any of the districts can be formed from the plurality of sectors include a restriction that, at the end of play, every sector within the region must be assigned to a district.

17. The method of claim 7, wherein the restrictions governing how any of the districts can be formed from the plurality of sectors include a restriction that each sector is assigned in its entirety to exactly one district.

18. The method of claim 7, wherein the step of making one move per turn per player comprises alternating moves between each player.

19. The method of claim 7, wherein the step of providing a region comprises providing a separate identical region to each player, and the step of making one move per turn per player comprises each player independently assigning a sector to a district within the region provided to that player.

20. The method of claim 7, wherein the step of making one move per turn comprises one of: (a) establishing a new district by assigning the first sector that belongs to it; (b) expanding a district by assigning an unassigned sector to an already established district; (c) reassign a sector from one district to another district, (d) breaking up one or more adjacent districts by un-assigning all sectors that belong to these adjacent districts; and (e) freezing a district by prohibiting any player from modifying the district during the next one or more turns.

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