WO2015070138A1 - Puzzle and game system and method - Google Patents

Abstract

In various embodiments, a system usable by a method to form at least one object based on calibrated degrees of difficulty includes a shape set comprising a first group and a second group. The first group includes a plurality of first shapes, and the second group includes a plurality of second shapes. The shape set includes a first combination set that forms a first object using first shapes and second shapes. Each combination in the first combination set has a first quantity of solutions that form the first object. The first quantity is greater than one and corresponds to a first degree of difficulty.

Description

PUZZLE AND GAME SYSTEM AND METHOD

CROSS REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. provisional application serial no.

61902512, filed on November 11, 2013, and U.S. patent application no. 14/080,617, filed on November 14, 2013, the entirety of each of which is incorporated herein by reference. BACKGROUND

Technical Field

[0002] This invention relates to puzzles, and more particularly to a puzzle system and method.

Description of the Related Art

[0003] Conventional games may provide several pieces for forming a shape, but may lack sufficient depth to hold a player's interest for very long. For example, in the simplest type of puzzle, once the puzzle has been completed, the players have no reason to use the puzzle a second time. In some other conventional systems, several pieces may be provided that can be combined in different ways to form a shape. In these other conventional systems, however, the player may still have little reason to choose one particular puzzle over another, which may lead to a lack of player interest. Accordingly, improvements may be made over conventional puzzles and puzzle games.

SUMMARY

[0004] In an aspect, a system to form at least one 3D object based on calibrated degrees of difficulty includes a shape set comprising a first group and a second group, the first group including a plurality of first 3D shapes, and the second group including a plurality of second 3D shapes. The shape set includes a first combination set that forms a first object using first shapes and second shapes, each combination in the first combination set having a first quantity of solutions that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty. In some embodiments, the first combination set uses only first shapes and second shapes.

[0005] The shape set may include a third group that includes at least one third shape. [0006] The shape set may include a second combination set that forms the first object using first shapes and second shapes, each of the combinations in the second combination set having a second quantity of solutions that form the first object, the second quantity corresponding to a second degree of difficulty. In some embodiments, the shape set consists only of first shapes and second shapes. The first quantity of solutions may be lower than the second quantity of solutions, and the second degree of difficulty is lower than the first degree of difficulty. Each of the first combination set and the second combination set may include at least two different solutions, or at least five different solutions.

[0007] The shape set may include a third combination set that forms a second object using first shapes and second shapes, each of the combinations in the third combination set having a third quantity of solutions that form the second object, the third quantity corresponding to a third degree of difficulty. The first quantity of solutions may be lower than the second quantity of solutions, and the second degree of difficulty may be lower than the first degree of difficulty. The second quantity of solutions may be lower than the third quantity of solutions, and the third degree of difficulty may be lower than the second degree of difficulty.

[0008] Each first shape may have a first volume, and each second shape may have a second volume. Each first shape may be equivalent to a first structure composed of five unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the first structure. Each second shape may be equivalent to a second structure composed of four unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the second structure. Each combination of the first combination set may use three first shapes and three second shapes, and each of the first shapes may be nonplanar.

[0009] In another aspect, a method of forming at least one object based on calibrated degrees of difficulty uses a shape set that includes a first group and a second group. The first group includes a plurality of first shapes, and the second group includes a plurality of second shapes, the shape set including a first combination set that forms a first object using first shapes and second shapes, each combination in the first combination set having a first quantity of solutions that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty.

[0010] The method includes providing the first combination set to a player. The first combination set includes a plurality of combinations that form the first object using first shapes and second shapes. Each of the combinations within the first combination set has the first quantity of solutions that form the first object. The first quantity is greater than one.

[0011] The method includes providing the first shapes that are used in the combinations of the first combination set, and providing the second shapes that are used in the combinations of the second combination set. The shape set may include a second combination set that forms the first object using first shapes and second shapes. Each of the combinations in the second combination set may have a second quantity of solutions that form the first object, and the second quantity may correspond to a second degree of difficulty. The first quantity of solutions may be lower than the second quantity of solutions, and the second degree of difficulty may be lower than the first degree of difficulty. Each of the first combination set and the second combination set may include at least two different solutions.

[0012] Each first shape may be equivalent to a first structure composed of exactly five unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the first structure. Each second shape may be equivalent to a second structure composed of exactly four unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the second structure.

[0013] In an additional aspect, a computer based method is provided for forming at least one object based on calibrated degrees of difficulty using a shape set that includes a first group and a second group. The first group includes a plurality of first shapes, and the second group includes a plurality of second shapes. The shape set includes a first combination set that forms a first object using first shapes and second shapes. Each combination in the first combination set has a first quantity of solutions that is greater than one, that corresponds to a first degree of difficulty, and that forms the first object.

[0014] The method includes, using a processing device and a display, providing the first combination set to a player. The first combination set includes a plurality of solutions that form the first object using first shapes and second shapes, each of the combination within the first combination set having the first quantity of solutions that form the first object, the first quantity being greater than one. The method also includes providing the first shapes that are used in the combinations of the first combination set, and providing the second shapes that are used in the combinations of the second combination set.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Fig. 1 is a block diagram of a puzzle system, according to an embodiment;

[0016] Fig. 2 is a flow chart of a puzzle method, according to an embodiment;

[0017] Figs. 3A-3C are illustrations of example objects that may be formed, according to several embodiments;

[0018] Figs. 4A-4B, 5A-5B, 6A-6B, 7A-7B, 8A-8B, 9A-9B, 10A-10B, 11A-11B, 12A- 12B, 13A-13B, and 14A-14D are illustrations of shapes that may be included in the puzzle system, according to an embodiment;

[0019] Fig. 15 is a block diagram of an exemplary computing system; and

[0020] Fig. 16 is a block diagram of a user computing device in communication with a server via a network, according to an embodiment.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0021] Exemplary embodiments of the invention are described in detail below. Note that the following exemplary embodiments do not in any way limit the scope of the invention. Note also that all of the elements described in connection with the following exemplary embodiments should not necessarily be taken as essential elements of the invention. I. Detailed Description of Exemplary System Components and Method Operations

[0022] In various embodiments, a puzzle system and method with calibrated difficulty levels is provided that allows players to select a puzzle with a level of difficulty that suits their own desire to challenge themselves. For example, the puzzle may include several combinations of different shapes that have the same difficulty level, which can allow a player to try several different puzzle combinations at the same or similar difficulty level before moving to another difficulty level. This can enhance replayability using the system by providing calibrated difficulty levels for different puzzles. [0023] In other embodiments, the puzzle system and method involves a multiplayer game that allows two or more people to race to solve a puzzle, with each player being able to select a puzzle of a particular difficulty level that corresponds to their ability. For example, a parent may compete with a child, with the parent selecting a puzzle with a higher degree of difficulty, and the child being given a puzzle with a lower degree of difficulty.

[0024] A shape set may be provided as part of the system that includes two to three different types of shapes that have been selected to provide varying quantities of solutions and corresponding difficulty levels. The more solutions that are available for a given shape, the easier it may be to find one of the solutions.

[0025] The objects to be formed with calibrated difficulty levels may include a 3x3x3 cube, a 3x3x4 block, and a 4x4x4 cube, each being equivalent to structures (e.g., polycubes) formed from unit cubes (e.g., a cube with a unit length measurement for each of length, width, and height). In other embodiments, additional shapes may be formed.

A. System Overview

[0026] Fig. 1 is a block diagram of a puzzle system, according to an embodiment. As shown, Fig. 1 includes a shape set 102, a first group 104, a second group 106, a third group 108, first shapes 1 10, second shapes 112, a third shape 114, first object

combinations 116, a 1st combination set 118, a 2nd combination set 120, a 3rd combination set 122, an nth combination set 124, Lth object combinations 126, a (N+l)th combination set 128, a (N+2)th combination set 130, a (N+M)th combination set 132.

[0027] The shape set 102 may include each of the shapes illustrated in Figs. 4A-14B, and may be organized into three groups: the first group 104 that includes the first shapes 110; the second group 106 that includes the second shapes 112; and the third group 108 that includes the third shape 1 14. The first group 104 may include shapes A through O (e.g., Figs. 4A-11A). The second group 106 may include shapes R through W (e.g., Figs. 11B- 14A). The third group 108 may include and/or consist of shape Z (e.g., Fig. 14B). Each shape may be equivalent to a structure that can be formed from unit-cubes (e.g., a cube with dimensions of one unit for each of length, width, and height), where each unit-cube of the structure is attached to at least one other unit-cube of the structure with the attached faces being aligned to match up against each other. One unit may be any length, such as 1.5 cm, 1.44 inches, 23 mm, etc. The first shapes 110, second shapes 112, and/or third shape 114 may be integrally formed and/or composed of unit cubes.

[0028] In other embodiments, the shape set 102 includes a fourth group that includes a fourth shape, and a fifth group that includes a fifth shape (not shown). The fourth group may include shape X, and the fifth group may include shape Y.

[0029] The first shapes 110 may be equivalent to structures formed from five unit-cubes that are attached with one full face matched against the full face of another unit cube, and may be non-planar (e.g., not capable of having all unit cubes contact the same planar surface, or having a dimension greater than one unit for each of length, width, and height). The first shapes 110 may not have any planar shapes.

[0030] The second shapes 112 may be equivalent to structures formed from four unit- cubes that are attached with one full face matched against the full face of another unit cube, and may be non-linear (e.g., having a dimension greater than one unit for at least two of length, width, and height). The second shapes 112 may lack a structure equivalent to a planar 2x2 square formed from unit cubes.

[0031] The third shape 114 may be equivalent to structures formed from three unit-cubes that are attached with one full face matched against the full face of another unit cube, may be non-linear, and may be limited to a single shape.

[0032] The fourth shape may be equivalent to structures formed from one unit-cube, and may be limited to a single shape.

[0033] The fifth shape may be equivalent to structures formed from two unit-cubes, that are attached with one full face matched against the full face of another unit cube, and may be limited to a single shape.

[0034] Each of the first shapes 1 10, second shapes 112, and third shape 114 may be capable of being disposed within a 3x3x3 cubic space.

[0035] As shown in Figs. 3A-3C, various combinations of shapes from the first group 104, second group 106, third group 108, fourth group, and/or fifth group may be used to create objects such as a 3x3x3 cube 302 (e.g., Fig. 3A), a 3x3x4 block 304 (e.g., Fig. 3B), or a 4x4x4 cube 306 (e.g., Fig. 3C). In other embodiments, other objects such as a hollow cube, cup, boat, chair, or a 5x5x5 cube (not shown) may be formed. [0036] As used in this description, a combination is a group of shapes (e.g., shape A, shape O, shape Z, etc.) that can be used to form a desired object (e.g., 302, 304, 306) with a calibrated difficulty level. The first object combinations each form the first object, and the Lth object combinations each form the Lth object.

[0037] Each combination set corresponds to a particular quantity of solutions (e.g., a specific arrangement of shapes that creates the desired object). In other words, each of the combinations in a particular combination set have the same quantity of solutions that will form the desired object. The quantity of solutions for a given combination may be 0, 1, 55, or even greater.

[0038] A combination with 0 solutions would be impossible to solve. A combination having only one solution may correspond to a puzzle that is relatively difficult to solve, given that fewer options exist for creating the desired object. Having more possible solutions may correspond to a puzzle that is easier to solve, because whether by chance or design, a player is more likely to come across one of the multiple solutions.

[0039] By providing multiple solutions having the same or similar difficulty, the system provides options for replaying the puzzle with the same or similar difficulty levels without using exactly the same pieces. In this way, multiple players can compete using different pieces while knowing that they were being challenged at the same or similar difficulty levels. Alternatively, one or more of the players may try to or at different levels to create handicaps for themselves.

[0040] Operations of a puzzle method that can be used with the system based on its calibrated difficulty levels are described in greater detail below.

B. Exemplary Combinations, Each Having One or More Solutions

1. First Object combinations

[0041] The first object combinations 116 may include one or more combination sets for forming a 3x3x3 cube 302. As shown, the first object combinations 116 include a 1st combination set 118, a 2nd combination set 120, a 3rd combination set 122, and an nth combination set 124. Here, n represents an integer greater than 4 (e.g., 37, 33, 17, 5, etc.), and the first object combinations may include combination sets 1 through n. Each of the combination shown includes six letters, which each represent a shape. The first three letters may represent a first shape 110 from the first group 104, and the second three letters may represent a second shape 112 from the second group 106.

[0042] Each of the solutions in the 1st combination set 118 may have XI solutions (e.g., 1 solution). In other words, there may be only XI ways to combine each of the pieces for a given solution to form the first object. For example, if XI represents 1, then pieces ABCRSV may be combined in only one way (e.g., one solution) to create a 3x3x3 cube. Other combinations included in the 1st combination set 118 may include ABCSUV, ABESTW, and ABESVW, each of which may be combined in only XI solutions to form the 3x3x3 cube.

[0043] The 2nd combination set 120 may have X2 (e.g., 3) solutions for each

combination. For example, if X2 = 3, the combination of shapes DFIRSW, DFIRTW, DFISUW, and ADJSVW may each be arranged in three different ways to create the first object 116 (e.g., the 3x3x3 cube). The 3rd combination set 122 may have X3 (e.g., 12) solutions for each combination. In other words, if X3=12, the combinations in the 3rd combination set 122 may each be combinable in exactly 12 different ways to create the first object 116. The nth combination set 124 may have XN (e.g., 24) solutions for each combination in the nth combination set 124 to create the first object 116.

[0044] Based on the exemplary number of solutions provided above, each of the solutions in the 3rd combination set 122 may be easier to solve than the solutions in the 2nd combination set 120, which in turn may be easier to solve than each of the solutions in the 1st combination set 118.

2. Lth Object combinations

[0045] As shown in Fig. 1, the system may also have Lth object combinations 126. Here, L may represent a number greater than 1, such as 2, 4, or higher numbers. Other objects between the first object and the Lth object may also be formed using the shape set 102.

[0046] The Lth object combinations 126 may represent the shapes that may be used to form a 4x4x4 cube 306. Twelve of the first shapes 110 from the first group 104 may be combined with one of the second shapes 112 from the second group 106. A solution may be represented as "-AFI+W" which can be read to mean all of the first shapes 110, with the exception of shapes A, F, and I, combined with shape W from the second group 106. This achieves a higher density of codes that may be needed for the puzzle and game system.

[0047] The Lth object combinations 126 may include the (N+l)th combination set 128, the (N+2)th combination set 130, and the (N+M)th combination set 132. M here represents an integer greater than 2, such as 3, 10, or higher numbers. Each combination in the (N+l)th combination set 128 may have Yl (e.g., 1) solutions that form the Lth object (e.g., the 4x4x4 cube). Each combination in the (N+2)th combination set 130 may have Y2 (e.g., 10) solutions that forms the Lth object, and each combination in the (N+M)th combination set 132 may have YM (e.g., 53) solutions that form the Lth object.

3. Other Object combinations

[0048] Other objects may be formed using the shape set 102, and/or other solutions using different shapes exist.

[0049] The first combination set 118 may include combinations that use shapes from each of the first group 104, the second group 106, and the third group 108, for forming the 3x3x3 cube 302. For example, the first combination set 118 that has XI (e.g., one) solution for each combination may include: ABCEWZ, BEINWZ, and EFHMUZ. The first object combinations 116 may also include ABDFTZ and CDHJTZ, which may each be a combination with 92 solutions.

[0050] The shape set 102 may further be used to form other objects, such as a 3x3x4 block 304. The second combination set 120 with three solutions for each combination may include BGLO-ST, JLMN-TV, and GILN-ST, where the first four letters designate first shapes 110 to include, and the minus symbol followed by two letters (e.g., "-st") indicates which second shapes 112 to exclude out of all of the second shapes 112 in the second group 106.

C. Exemplary Operations

[0051] As shown in Fig. 2, a puzzle method may include operations 202, 204, and 206.

[0052] In operation 202, a first combination set (e.g., a first combination set 118, a 3rd combination set 122, an (N+2)th combination set 124) is provided to a player. The combination set may be provided as written instructions on paper, displayed on a screen, or provided via a non-transitory storage medium such as a CD, DVD, Blu-ray disk, flash drive, or hard disk. For a physical game with wooden, plastic, or shapes made from other materials, the combination set would likely be included on material such as paper or cardboard with the tangible puzzle shapes, such as in the same box or in other packaging. For a computer game, the combination set would likely be provided on non-transitory storage medium and/or downloaded with the game to be displayed by a processor on a display such as a smartphone screen, tablet screen, or laptop screen.

[0053] The first combination set 118 includes a plurality of combinations that form the first object (e.g., 302, 304, 306) using first shapes 110 and second shapes 1 12, each of the combinations within the first combination set 116 having the first quantity (e.g., XI, X2, XN) of solutions that form the first object, the first quantity being greater than one. As shown in Fig. 1, each of the exemplary combination sets includes at least one possible combination. Each combination in each combination set has an associated quantity of solutions that corresponds to the combination set and/or to a level of difficulty.

[0054] As discussed above, the method involves forming at least one object based on calibrated degrees of difficulty using a shape set 102 that includes the first group 104 and the second group 106. The first group 104 includes a plurality of first shapes 110, and the second group 106 includes a plurality of second shapes 112. In other embodiments, the shape set 102 may also include the third group 108 that includes one or more third shapes 114.

[0055] The shape set 102 includes a first combination set (e.g., 1st combination set 118, 3rd combination set 122, (N+M)th combination set 132) that forms a first object (e.g., the 3x3x3 cube 302, the 3x3x4 block 304, the 4x4x4 cube 306, etc.) using first shapes 110 and second shapes 112. In other embodiments, the third shape 114 may also be used in various combination sets.

[0056] Each combination in the first combination set has a first quantity of solutions (e.g., 1, 3, 12, 24, 10, 53, etc.) that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty,

[0057] In operation 204, the first shapes 110 that are used in the solutions of the first combination set are provided. For a physical game, the shapes may be provided by being included in a game set that includes the shape set 102. For a computer game, the first shapes 110 may be provided by a program that allows the user to access, use, and/or manipulate the first shapes 110 so that the user can solve the puzzle and form the desired object.

[0058] In operation 206, the second shapes 112 used in the combinations of the second combination set are provided.

[0059] In some embodiments, a two player game may be played using the system described herein. For example, Player 1 may select a code for player 2, and player 2 may select a code for player 1, not using the first three letters of player l's code. When both players are ready, player 1 and player 2 race to complete the desired object, such as a 3x3x3 cube 302. The first player to finish wins. Players can agree on a handicap using the solution levels (e.g., levels 1-59).

[0060] In some versions where the shape set includes one set of shapes A through O, and two sets of shapes R through W, Player 2 can select a combination that does not use the first three letters of Player l 's combination. In other versions, the shape set may include two sets of shapes A through O, and one set of shapes R through W, and Player 2 can select a combination that does not use the last three letters of Player 1 ' s combination. In other versions, each of Player 1 and Player 2 would use their own full set of shapes A through O and R through W, making it unnecessary to have restrictions on which letter shapes may be used. The shape sets may or may not include shape Z. Shapes X and Y may be used to create additional handicaps for beginners.

D. Detailed Description of Exemplary Shapes

[0061] Fig 4A-14D are illustrations of shapes that may be included in the puzzle system, according to an embodiment. For individual unit cubes, particular faces of the unit cubes may be identified by the direction (e.g., directions 1, 2, 3, 4, 5, and 6) that projects outwardly from the face of the unit cube. For example, the #6 face of a unit cube in the illustrations would be on the bottom, and the #5 face would be on the top.

[0062] Fig. 4A is an illustration of shape A, which can be equivalent to a structure constructed using five unit cubes A01, A02, A03, A04, and A05 that are assembled such that each unit cube of the structure has at least one face that is matched up with a face of another unit cube of the structure.

[0063] The structure of shape A can be formed by attaching the #6 face of unit cube A01 to the #5 face of unit cube A02. The #3 face of unit cube A02 is attached to the #4 face of unit cube A03. The #2 face of unit cube A03 is attached to the #1 face of unit cube A05. The #3 face of unit cube A03 is attached to the #4 face of unit cube A04.

[0064] Fig. 4B is an illustration of shape B, which can be equivalent to a structure constructed using five unit cubes B01, B02, B03, B04, and B05. The structure of shape B can be formed by attaching the #1 face of unit cube B01 to the #2 face of unit cube B02. The #3 face of unit cube B02 is attached to the #4 face of unit cube B03. The #1 face of unit cube B03 is attached to the #2 face of unit cube B04. The #5 face of unit cube B04 is attached to the #6 face of unit cube B05.

[0065] Fig. 5A is an illustration of shape C, which can be equivalent to a structure constructed using five unit cubes COl, C02, C03, C04, and C05. The structure of shape C can be formed by attaching the #1 face of unit cube COl to the #2 face of unit cube C02. The #1 face of unit cube C02 is attached to the #2 face of unit cube C03. The #5 face of unit cube C03 is attached to the #6 face of unit cube C04. The #3 face of unit cube C04 is attached to the #4 face of unit cube C05.

[0066] Fig. 5B is an illustration of shape D, which can be equivalent to a structure constructed using five unit cubes D01, D02, D03, D04, and D05. The structure of shape D can be formed by attaching the #1 face of unit cube D01 to the #2 face of unit cube D02. The #1 face of unit cube D02 is attached to the #2 face of unit cube D03. The #6 face of unit cube D03 is attached to the #5 face of unit cube D05. The #3 face of unit cube D03 is attached to the #4 face of unit cube D04.

[0067] Fig. 6A is an illustration of shape E, which can be equivalent to a structure constructed using five unit cubes E01, E02, E03, E04, and E05. The structure of shape E can be formed by attaching the #6 face of unit cube E01 to the #5 face of unit cube E02. The #3 face of unit cube E02 is attached to the #4 face of unit cube E03. The #2 face of unit cube E03 is attached to the #1 face of unit cube E04. The #1 face of unit cube E03 is attached to the #2 face of unit cube E05.

[0068] Fig. 6B is an illustration of shape F, which can be equivalent to a structure constructed using five unit cubes F01, F02, F03, F04, and F05. The structure of shape F can be formed by attaching the #1 face of unit cube F01 to the #2 face of unit cube F02. The #1 face of unit cube F02 is attached to the #2 face of unit cube F03. The #3 face of unit cube F03 is attached to the #4 face of unit cube F04. The #5 face of unit cube F04 is attached to the #6 face of unit cube F05.

[0069] Fig. 7 A is an illustration of shape G, which can be equivalent to a structure constructed using five unit cubes G01, G02, G03, G04, and G05. The structure of shape G can be formed by attaching the #3 face of unit cube G01 to the #4 face of unit cube G02. The #1 face of unit cube G02 is attached to the #2 face of unit cube G03. The #6 face of unit cube G03 is attached to the #5 face of unit cube G05. The #3 face of unit cube G03 is attached to the #4 face of unit cube G04.

[0070] Fig. 7B is an illustration of shape H, which can be equivalent to a structure constructed using five unit cubes H01, H02, H03, H04, and H05. The structure of shape H can be formed by attaching the #4 face of unit cube H01 to the #3 face of unit cube H02. The #1 face of unit cube H02 is attached to the #2 face of unit cube H03. The #5 face of unit cube H03 is attached to the #6 face of unit cube H05. The #1 face of unit cube H03 is attached to the #2 face of unit cube H04.

[0071] Fig. 8 A is an illustration of shape I, which can be equivalent to a structure constructed using five unit cubes 101, 102, 103, 104, and 105. The structure of shape I can be formed by attaching the #1 face of unit cube 101 to the #2 face of unit cube 102. The #6 face of unit cube 102 is attached to the #5 face of unit cube 105. The #3 face of unit cube 102 is attached to the #4 face of unit cube 103. The #1 face of unit cube 103 is attached to the #2 face of unit cube 104.

[0072] Fig. 8B is an illustration of shape J, which can be equivalent to a structure constructed using five unit cubes J01, J02, J03, J04, and J05. The structure of shape J can be formed by attaching the #6 face of unit cube J01 to the #5 face of unit cube J02. The #3 face of unit cube J02 is attached to the #4 face of unit cube J03. The #1 face of unit cube J03 is attached to the #2 face of unit cube J04. The #3 face of unit cube J04 is attached to the #4 face of unit cube JO 5.

[0073] Fig. 9A is an illustration of shape K, which can be equivalent to a structure constructed using five unit cubes K01, K02, K03, K04, and K05. The structure of shape K can be formed by attaching the #3 face of unit cube K01 to the #4 face of unit cube K02. The #1 face of unit cube K02 is attached to the #2 face of unit cube K03. The #5 face of unit cube K03 is attached to the #6 face of unit cube K04. The #3 face of unit cube K04 is attached to the #4 face of unit cube K05.

[0074] Fig. 9B is an illustration of shape L, which can be equivalent to a structure constructed using five unit cubes L01, L02, L03, L04, and L05. The structure of shape L can be formed by attaching the #1 face of unit cube L01 to the #2 face of unit cube L02. The #6 face of unit cube L02 is attached to the #1 face of unit cube L03. The #3 face of unit cube L03 is attached to the #4 face of unit cube L04. The #1 face of unit cube L04 is attached to the #2 face of unit cube L05.

[0075] Fig. 10A is an illustration of shape M, which can be equivalent to a structure constructed using five unit cubes M01, M02, M03, M04, and M05. The structure of shape M can be formed by attaching the #1 face of unit cube M01 to the #2 face of unit cube M02. The #1 face of unit cube M02 is attached to the #2 face of unit cube M03. The #3 face of unit cube M02 is attached to the #4 face of unit cube M04. The #6 face of unit cube M02 is attached to the #5 face of unit cube M05.

[0076] Fig. 10B is an illustration of shape N, which can be equivalent to a structure constructed using five unit cubes N01, N02, N03, N04, and N05. The structure of shape N can be formed by attaching the #6 face of unit cube N01 to the #5 face of unit cube N02. The #1 face of unit cube N02 is attached to the #2 face of unit cube N03. The #1 face of unit cube N03 is attached to the #2 face of unit cube N04. The #4 face of unit cube N04 is attached to the #3 face of unit cube N05.

[0077] Fig. 11 A is an illustration of shape O, which can be equivalent to a structure constructed using five unit cubes O01, O02, O03, O04, and O05. The structure of shape O can be formed by attaching the #1 face of unit cube O01 to the #2 face of unit cube O02. The #3 face of unit cube O02 is attached to the #4 face of unit cube O03. The #3 face of unit cube O03 is attached to the #4 face of unit cube O04. The #5 face of unit cube O04 is attached to the #6 face of unit cube O05.

[0078] Fig. 1 IB is an illustration of shape R, which can be equivalent to a structure constructed using five unit cubes ROl, R02, R03, and R04. The structure of shape R can be formed by attaching the #3 face of unit cube ROl to the #4 face of unit cube R02. The #1 face of unit cube R02 is attached to the #2 face of unit cube R03. The #5 face of unit cube R03 is attached to the #6 face of unit cube R04. [0079] Fig. 12A is an illustration of shape S, which can be equivalent to a structure constructed using five unit cubes S01, S02, S03, and S04. The structure of shape S can be formed by attaching the #1 face of unit cube S01 to the #2 face of unit cube S02. The #3 face of unit cube S02 is attached to the #4 face of unit cube S03. The #5 face of unit cube S03 is attached to the #6 face of unit cube S04.

[0080] Fig. 12B is an illustration of shape T, which can be equivalent to a structure constructed using five unit cubes T01, T02, T03, and T04. The structure of shape T can be formed by attaching the #1 face of unit cube T01 to the #2 face of unit cube T02. The #1 face of unit cube T02 is attached to the #2 face of unit cube T03. The #5 face of unit cube T03 is attached to the #6 face of unit cube T04.

[0081] Fig. 13 A is an illustration of shape U, which can be equivalent to a structure constructed using five unit cubes U01, U02, U03, and U04. The structure of shape U can be formed by attaching the #1 face of unit cube U01 to the #2 face of unit cube U02. The #1 face of unit cube U02 is attached to the #2 face of unit cube U04. The #5 face of unit cube U02 is attached to the #6 face of unit cube U03.

[0082] Fig. 13B is an illustration of shape V, which can be equivalent to a structure constructed using five unit cubes V01, V02, V03, and V04. The structure of shape V can be formed by attaching the #1 face of unit cube V01 to the #2 face of unit cube V02. The #5 face of unit cube V02 is attached to the #6 face of unit cube V03. The #1 face of unit cube V03 is attached to the #2 face of unit cube V04.

[0083] Fig. 14A is an illustration of shape W, which can be equivalent to a structure constructed using five unit cubes W01, W02, W03, W04. The structure of shape W can be formed by attaching the #3 face of unit cube W01 to the #4 face of unit cube W02. The #2 face of unit cube W02 is attached to the #1 face of unit cube W03. The #5 face of unit cube W02 is attached to the #6 face of unit cube W04.

[0084] Fig. 14B is an illustration of shape Z, which can be equivalent to a structure constructed using five unit cubes Z01, Z02, and Z03. The structure of shape Z can be formed by attaching the #1 face of unit cube Z01 to the #2 face of unit cube Z02. The #3 face of unit cube Z02 is attached to the #4 face of unit cube Z03.

[0085] Fig. 14C is an illustration of shape X, which can be equivalent to a structure constructed using one unit cubes X01. [0086] Fig. 14D is an illustration of shape Y, which can be equivalent to a structure constructed using two unit cubes Y01 and Y02. The structure of shape Y can be formed by attaching the #3 face of unit cube Y01 to the #4 face of unit cube Y02. E. Exemplary Combination Sets

[0087] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 1 solution for each combination includes the following combinations:

abcrsv abcsuv abestw abesvw abfrsv abfrsw abfsvw abgrsu abgrtv abgstu abgsuv abguvw abhrsu abhruv abiruw abisuv abiuvw abjsuv abkrtu abkrtv abkrtw abkstu abktvw ablrtv ablrtw ablstw abmstv abnrsv abnuvw aboruw acdsvw acersv.

[0088] The combination set may also include: abcewz abcirz abcisz abckvz abclwz abcmsz abegsz abejsz abektz abeltz abeowz abfjwz abfksz abghsz abgltz abgmsz abhkuz abhmrz abhmsz abiltz abimsz abjnvz abjosz abjovz abkmwz ablnrz ablnwz ablosz abmouz abmowz abnosz abnovz.

[0089] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 2 solutions for each combination includes the following combinations: clntuv agmstu adfrvw bdotvw befrsv befrsw clotuw befruv cmnrtu cmnrtw befstw abmrvw cmnstu begruw cmnsuw adhrvw cmorsv cmorsw cmorvw cmostv begtuw behrst agnsuv behrsw cnosvw cnotvw agorsu adhsvw defrvw defstw defsuw behruw.

[0090] The combination set may also include: bceguz bcehsz cehisz acehvz bcehwz aceirz bceiuz ceijsz aehjrz ceinwz aehjuz bcektz aehjvz cejmsz cejosz cejovz ceknsz bceosz celotz bceowz cemnvz cemosz cemovz aehltz cfgirz cfgisz cfgiwz cfgjrz cfgjuz cfgktz abfkrz.

[0091] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 3 solutions for each combination includes the following combinations:

dfirsw dfirtw dfisuw adjsvw fhjrtw bchrtw fhjruw fhjsuw fhjtvw fhkrst fhkrsw aclrsw fhkrtv behruw fhkstv cenruv bchstw fhktuw bgjrtu afosvw aimstw dfktuv dflrtv censvw dflruv dflstw ceorsu dfmrsw fhmstu ceorsw ceortv bchtuv. [0092] The combination set may also include: agnowz cenouz acfkuz bcgjtz dikovz bfhiuz dilnsz adejvz ahijuz abhjrz ahiktz aejktz bfhlwz bfhmrz abknwz djkorz aejmtz cfhkuz djlnsz ahjkrz djlorz djlosz djlowz cfhkwz djmowz cfhlrz djnosz cfhluz bfimuz acfluz cfhmuz.

[0093] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 4 solutions for each combination includes the following combinations: acistv fjmstv bdersw cimrtw bdertv afgtuw fjnrsv afguvw fjnruv afhrsv bgortu bdeuvw ahmtvw cinsuw bdfruw cintvw fjotuw fjotvw cdnrtw cdnruv bdfsuw ciostv bdfsvw fknrtv cdnsuw dlmstv fknstu akostv cdnuvw bhirtv adgstw cjkstu.

[0094] The combination set may also include: acejvz efijtz cikouz abfiwz aefltz ailntz adeksz bchmwz cdgisz afjlwz bchosz abdmrz abiowz bcimtz aefnuz cjkmtz efjosz abjktz bcinuz eflntz cdglvz afjmwz afjnrz abfjuz cjlnsz cdgnrz cdgnwz adfhwz efnowz cjlosz adfkvz.

[0095] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 5 solutions for each combination includes the following combinations: aegtuv dgkstu bgotuv bgotuw cgnrtv hijtvw fhnruv dglrst dglrtu afkrtu bknrtu dglstv acgrtv ajnstw dgmrtw bhirtu hinrst bknstu adktvw abktuv fhosuv fhosuw fhotuw aijtvw dgnstw dgnsuv cehtvw dgnuvw efortv cgostv dgorsu.

[0096] The combination set may also include: cfijrz cdelvz dhlnrz adgnwz aflosz bfnosz fhinwz adgorz fhiouz bcknrz dhlouz cdenwz dhmnuz defisz cdeorz dhmouz defjrz defjsz dhnouz ahmntz adgosz dijktz cfinsz fhkntz ahmotz defkvz adgowz cfiovz acghrz adhkvz bcdovz.

[0097] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 6 solutions for each combination includes the following combinations: fiostu ehnrtv djkrtw hiouvw bjnruw cinstw aeostv bentvw cdltuw bjntuv deorsu fjlrtu bjntuw djlrtw deortw hjlrtv ehostu ahoruv abcrtw adisuv cdmstv djmrsv abnsuw ehotvw ehouvw beortv cdmtuv hjmrtu cfntuw cjkrst cjkrtu.

[0098] The combination set may also include: agjnuz chmowz bdhmuz fhnosz abchsz aekntz dfiorz chnowz cfgnwz agjouz cfgouz bdhnsz bcdmsz adinuz cfhivz dfjmrz adiowz bhinuz cijouz cdgnsz aekotz bhiouz fikotz benotz abdjwz afiovz filotz adeosz abenwz bdimsz bfgitz. [0099] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 7 solutions for each combination includes the following combinations: admsvw afjtuw bfmstw ahirst hnorsu bfnrsu bfnrtv afnstu cijrtu dhistw fjortv fgmtuv cgjtuv bdntvw cfgstu cijstu abduvw ajmrtv bmortv cijtuv fgntuw dfgruw dhjuvw afgstu abmstw acnuvw ahistw fgostv cfhrsv bfgrtu cfhruv.

[00100] The combination set may also include: cdiorz cdiowz abhorz bchjrz abdnvz ablntz abhouz afgovz fgjosz efgntz cenosz abcjuz begntz ahiorz cdjnrz cfghuz bdegtz cdjovz ghmntz cdjowz aceorz aghnsz adjkrz fgmntz cjmorz cjmosz afhiwz dfgnsz dfgnwz fgnovz efhnsz.

[00101] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 8 solutions for each combination includes the following combinations: fnorst fhotuv acotvw acituv biotuw abhuvw dfktvw himtuv dflrst dehstw bfistv adersv cdirtw cjotvw aintvw afhtuw bfjrst adertw bdfrvw hintuv hintuw deistv dfmrvw adnruv dejrst fghstv cdituv fghtuv abortv dhnruv hiostv.

[00102] The combination set may also include: acjouz dhjlrz fgnorz adfluz cefnsz cfinrz cefnvz dfimtz ceforz cefovz dhjnwz bginuz cfiouz ceghtz cdgmtz fhinsz aefitz acjovz bcdkvz agnotz gijouz bcdltz dfjktz ginouz begnvz dhkotz adhmvz acnovz bfhmtz bdfiwz dfjlvz.

[00103] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 9 solutions for each combination includes the following combinations: abituw chnstw bdnstv dghstu fhjstv abirtv efntvw afortw benrtv aknrtv fjosuv dhouvw ejostv afktuw cnorsw cnortu cjmtuw dlmrtv bdorsu gintuw afhstu bgnuvw cnostw dijuvw afostw giotuw dgjrst cfktuv cgnstv bdguvw inortu.

[00104] The combination set may also include: eghntz adjmrz achnrz adekvz bcfivz adinrz cdjorz bcdnsz dfjnsz fjmnsz chjorz cdfnvz dfjorz adfiuz cdfovz cdghuz bcdorz bdhkvz cghnrz ckmotz dejnrz cghnuz bdeosz afiouz adkovz ehintz aehntz aefnsz cehnsz cdkovz adnotz.

[00105] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 10 solutions for each combination includes the following combinations: adhruv cgjstv cflstu bfkrtu bfntuv fnostv dnotuv acotuw dfnsvw hmortw ajotuv adfruv fhnstv gjouvw cdgtuv ejnrtv hmotvw bejtvw fjntuv cdmrtw fhntuw fjorst fjorsu beftuw afhrtu amnstw dfgtvw adfruw bcdstv bjmtvw fgnrst.

[00106] The combination set may also include: bfimtz abdmsz befjuz fgmotz bdfmrz abfovz abfowz fgnosz fgnotz bcefrz dhjktz abcnuz abejtz adhmtz bcfovz acjnsz aciktz dhjmrz abkntz adlnvz cfjnvz cdijrz agjntz bdjmwz cikntz bfjmtz dhjowz bdegsz acjorz acjosz bfjnuz cfkorz.

[00107] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 11 solutions for each combination includes the following combinations: bgituv ciortu afituv bmorst fjortu cfjtvw ciostu bdhtuw adgsuw afktuv dghrtw cdgtvw bentuv fgnrtu adnsuv hnostv adistw dfitvw fgnstu aenrst cghtuv ejotuv aghrtv bdnrtu amotvw dkostv dhktvw cdmstw jmnrst fkortu abcrtu.

[00108] The combination set may also include: cejouz ehnotz efintz afgosz cfhnvz adfouz bcfhwz aenotz adfovz cdkntz cdhkrz abdkuz cdenvz dhkntz bcefuz fhiotz dfjmtz bcdeuz cdeovz abfnuz ahjovz adlotz acfjwz cdhnuz cdhnvz chinrz achjvz cegotz bcdhsz bdefrz cknotz.

[00109] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 12 solutions for each combination includes the following combinations: dforst bdjuvw ahnrtv cdirtu aiorst dgostv adktuv bijtuv djmstv afjrtv dhotuv dfistv afgrst bjnstu acjstu cijtuw acnrtv ehjtuw hiotvw afnstv bcorsu adhrsu bfgtuw bchstv ghnrst bdhrst abhstv bjortu cdfrsv djortw cdfrtw.

[00110] The combination set may also include: abdorz achjrz dhmntz cdfluz bfhktz cfgnrz fgjntz cdiktz adenuz adfosz ahjmtz aciosz dfgltz abdnrz acdkuz bfkntz cdfnwz bcnouz adejrz cdfowz acdguz bmnotz dfgnuz dgmovz dfgnvz cdinuz abcosz acfowz dflntz dfgouz dehmtz.

[00111] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 13 solutions for each combination includes the following combinations:

hjortu hjortv dfjsuv cdjrsu chjrtu dhistu hjostv djnstw adiuvw fhorst bcostu cfnrtv fjnrtu bijtuw fgorst cjortw fjnstu ijortu bdjruv anortw dghtuw afnrtw dghuvw djoruv dehstv ijotuw dhjstw chjtvw ceortu bdjsuv dhkrst. [00112] The combination set may also include: aefnrz cdjmrz acdgsz dghmtz cfmntz afjltz bdfhrz acfnuz acdorz aejntz cfmotz bcjnuz befitz dghovz fhijtz hjmotz adimtz dehjrz cfhouz cdeouz efiotz cghitz bcfnsz ghnotz djlotz behotz cdfgsz afhorz cdfgvz abcfuz cdfirz.

[00113] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 14 solutions for each combination includes the following combinations:

cfortv adgtvw bdntuv inotvw chostv abnruv cdnstw afirtu dhitvw cjostw djotuv acostv bdjtuv fhnrtv cehstv dfltuv hjosuv ehjtuv dgituw bdfstv dfortw bdortv chnrtv cdftuv dfgtuw cdktuv acistw abgtuv cdituw cfnstv cghstu abfrtw.

[00114] The combination set may also include: bdfmtz ceijtz gjnotz acfltz binotz abeotz acdhvz bgmntz dhjorz cdegtz cfjnuz achjsz fhintz abfhrz dfhorz cdhnsz ghiotz abjouz achjwz cdhorz hijntz acfirz acdiuz abdjrz cfhjrz cefnrz cfhjvz abgotz adgmtz cfhktz cdghrz.

[00115] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 15 solutions for each combination includes the following combinations:

aimtuv dknrtv bcfuvw abjrtu adhstu adostv centvw bdhstv achrtv cfirst abituv dfjrtw bgntuv bhjstu ijotuv dfjstu chjrtw ghjtuv acfstw dfgrtu anostw dgjtuw adntvw cfistw adhrtu djnstu fjotuv dhnrtv abdruv bfituw adistu.

[00116] The combination set may also include: cdhmwz bcdouz abcktz defgtz hjlntz adfmwz fhjltz cfgotz bcfmtz cdhnwz bcejtz cdhowz abhnuz cfintz chkntz bdfltz afjntz adfnwz cdeltz cdjnsz defktz adenvz aefnrz ahjouz adhnsz agiotz acdnvz cdfnsz beghtz adhorz adeitz.

[00117] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 16 solutions for each combination includes the following combinations:

cgjstu hjotuv behtuw admrtw imotvw dfjrsu beftuv bdhrtv cdorst acfstv gmntvw dgotuv acjrtu fjortw adgrtv bfirtu afnrtu chjtuv djmrst bdmtuv abctuv adhtuv dhistv bdgrtu dfnrst dintuv djmtuv bdjrtw befrtw abntuv abftuv. [00118] The combination set may also include: gmnotz bdjmvz dgnotz cdhktz bdfguz acdnuz aehjtz acfouz adhiwz aghnvz afmntz imnotz cdfhvz cdfhwz cdefvz bcdhvz admnsz cefnuz chjnrz gjmntz dhmorz abfosz adfjvz abehtz cfhnuz adghwz dfhouz fjlotz cefouz bimotz chmotz.

[00119] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 17 solutions for each combination includes the following combinations: cdkrtv abdtvw dhortv abdstw adfrtw bdmtuw bcotuw adostw djmtuw dflstv adnrsu abdrsu abjtuv dhjrsu acjtuw gjnstu dhmrtw bdertu dhjrtw efntuv bcnrtu bdotuv bdnstu dfostu flntvw fjntuw djntuv fiortu hjostu dfjstw bcfstu cdnrtu.

[00120] The combination set may also include: chiotz adfnvz behntz cjlntz adgnuz cdehvz chjktz bfkotz dhiouz adhosz cdhosz cdhovz bdmnvz bdgouz cfjosz bcdhrz bcfnrz djmovz admosz acderz adfguz dhmnrz defhsz abdhsz adefvz bdhnuz cdhiuz cdhjwz cdfisz afgntz abfltz.

[00121] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 18 solutions for each combination includes the following combinations:

dlnstv defrtv fhjrtu dfjtuv fhnrtu ghorst dfotuw abfstv cdntuw bdnrtv bdnrtw chjrtv bdfsuv dfkrtu cfnrtu hjnrtu ceftuv cdjruv bcdtuv dfnrtv bdirtu bcftuw abcstu dkortv cdostv acostu cdlstu abostu efhrst adhuvw cfhtuw.

[00122] The combination set may also include: adektz afintz afjotz bejntz dghnvz adiovz dfgitz cfjnsz bcforz acfktz cfhltz bcdktz dfgjuz cfgntz acdeuz bejotz afhjtz fghjtz dejovz adhnwz abehtz cghotz cdehrz cfiotz adegtz dehitz edgitz behntz dfjltz abcitz adhowz.

[00123] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 19 solutions for each combination includes the following combinations:

efnrtu abfrst bfjstu censtu chorst bcjrtu afnrst ceostu chjrst efortu.

[00124] The combination set may also include: adintz abghtz acdfsz bdijtz abeduz bdgjtz hjmntz adfirz edghsz fghotz cdfhrz adghvz dehotz cfhjwz adhivz bedjuz bfjntz cfkotz defhrz ahijtz bcfgtz dghotz bcfltz cfiorz dfghvz dfhltz acdivz bhjntz dfmntz cflntz cdjouz. [00125] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 20 solutions for each combination includes the following combinations:

dhituw adfstu bcfrst dfgrtv cdetuv dgituv adftuv djnrtu deftuv cdhtuv cdhrtu dnotvw cfjrst dhirtv dimtvw acfrtw flostu adgstv adgtuw bdostu dfntuw cfhstw cdotuw cknrtu dgmtvw.

[00126] The combination set may also include: cjkotz bdhntz cgjntz adfgwz adhjsz bceotz bfhitz cdfmtz bcdgtz acfhrz efjntz acfhsz aciotz cejntz bflntz acdjvz bcdhuz afgotz cdjnuz adjotz dfijtz fghntz adhmwz efhntz chintz bdfouz abdhrz dmnovz abhotz acgjtz efnotz.

[00127] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 21 solutions for each combination includes the following combinations:

admtvw cfjrtw chjstv djmrtv aderst dhnstv bdmstv bdgtuw dfirst dejrtv cdgrst bcfstw dijtuw adetuv dehtuv cfntuv adortv cfotuv dfhrtv dhmtvw dfostv cdlstv acdstv dehrst bdestv dfkrtv cdnrtv abfrtv.

[00128] The combination set may also include: aijotz adfivz acdjrz cfhjsz dfhmtz bghntz dejotz acdmtz cghntz abcfrz cdghvz dfhjrz adehrz adehsz afiotz chnotz acdjuz abcdsz afnotz bdfhsz bdfhuz.

[00129] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 22 solutions for each combination includes the following combinations:

dfnstu dhnstw abdstv cdortu adortw dinstu cfirst dhjrtv aehtuv dgortu efostu acfrtu deostv cfhstu dfnrtu cdnstu denrtv cenrtu dfortu cdostu cfkrst.

[00130] The combination set may also include: dfhnuz adefuz cdfosz admntz dijntz adhjvz cdintz dfiotz aghotz dhjnuz dfhjsz adfhuz bckotz acdouz bcditz cdehuz adgntz bdejtz acdhuz dhiotz dhmosz dhmotz bdejvz cdgntz dgjouz bdinuz ahintz cdfnrz fjlntz bdgotz dmnotz.

[00131] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 23 solutions for each combination includes the following combinations: dhostw dfirtu ahorst dinrtu bcfrtv adnrtw adnstw dgostu cfostw bdfrtw cfnrtw cfjstv dghrtv defrst ahnrst dhortw adistv cdjstw cderst.

[00132] The combination set may also include: cijotz adghrz adghsz adhirz bfgntz adhisz dfintz cfjntz adeotz dehjuz dehntz achktz abfjtz bcnotz abdeuz afhltz fjnotz bchjtz bcfotz.

[00133] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 24 solutions for each combination includes the following combinations:

dejtuv cefrtu bdetuv dhjrst abdrst bdjtvw cdjrtw bdfstw cefstu.

[00134] The combination set may also include: bdnotz abdiuz acdfrz cdefuz adfktz djmntz cehotz cdhltz dgmotz efhjtz acghtz dimntz bdmotz dhjntz aefntz cdfhsz djnotz abcetz dghjuz abdotz afhitz.

[00135] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 25 solutions for each combination includes the following combinations:

ghnstv dmorst cfhtuv adirtv dmnrst acdrtv dghstv dfortv aiortv adfstv dfhstv adjtuv cdhrtv bdhtuv cdnstv acftuv.

[00136] The combination set may also include: dgmnvz chjltz bdgmtz ahiotz dhjouz abfktz acdhrz abdfuz abdhuz cfgjtz dimovz fgjotz bcintz abdnuz adfhsz dgjntz cdhjuz aghntz adhjuz dijmtz bcfitz bdehuz adejuz.

[00137] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 26 solutions for each combination includes the following combinations:

abdstu dgntuv acfstu bfortu cdortv dfnstv dehstu cfhrtu adestu bcdstu dfjrtu cjnstu dehrtu dhjrtu diotuv adertu.

[00138] The combination set may also include: abdfvz defntz bcentz acijtz bfghtz cdfjvz cfjltz dhiovz cdeotz efjotz adgnvz bdiouz cdhjvz dgjnuz bcfktz bcdfvz.

[00139] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 27 solutions for each combination includes the following combinations:

bdjstu bdgstu bdjrtu dgnstu diortu abfrtu chjstu dijrtu. [00140] The combination set may also include: fhnotz abcdvz dgjmtz dfghrz bdimtz dghntz ahjotz bdhmtz acdisz adjmtz acnotz abhntz adiotz dfhjvz.

[00141] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 28 solutions for each combination includes the following combinations:

bdfrst acdtuv bdjtuw dfhtuv cdjrtu dhirst bdfstu cdjtuv adgrst bdftuv cdjrst.

[00142] The combination set may also include: cdefsz bfhjtz bdjntz abcjtz adhnvz bdjmtz befntz adfgvz befhtz acejtz bdjotz adgotz adfgrz adhovz cdhisz dhintz cejotz cdhivz.

[00143] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 29 solutions for each combination includes the following combinations:

cdirst abdrtv bdistu abcrst adjstu bdhrtu dfgrst defstu cdertu dhjstv fhjrst dgjrtu.

[00144] The combination set may also include: dhjotz achitz cdentz bfhotz bcdetz abdntz dhjmtz achntz cdfhuz afhotz defotz bdfktz acjntz acdfuz afghtz abdmtz cdjltz.

[00145] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 30 solutions for each combination includes the following combinations:

cdirtv cdgstv dfgstv dfirtv.

[00146] The combination set may also include: dijouz defhtz chijtz abfgtz cdfjuz cfhitz cdjntz acfgtz bdgnuz acdetz cfhnsz cdiotz bdfotz dfgntz acforz bcdfuz.

[00147] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 31 solutions for each combination includes the following combinations:

adhstv adhrtv.

[00148] The combination set may also include: bdintz cfhotz ghjotz dgintz adfhrz dgiotz acdhsz adfgsz abintz cghjtz dgjotz acfntz abfitz cdhirz.

[00149] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 32 solutions for each combination includes the following combinations:

dhjstu cfhrst cdjtuw acdstu bdftuw abdrtu acfrst dfhrst adfrtu acdrst dfhrtu cdhstu. [00150] The combination set may also include: abdktz cdhmtz bcdntz adfmtz dhjltz dfjotz.

[00151] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 33 solutions for each combination includes the following combinations:

cdfstv cdfrtv.

[00152] The combination set may also include: dfghtz achjtz abfhtz acditz.

[00153] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 34 solutions for each combination includes the following combinations:

cdfrtu cdfstu.

[00154] The combination set may also include: aefhtz bcdfsz acjotz fhjotz adgjtz hijotz abgntz dimotz bdhitz abcotz bdgntz dijotz dgmntz abdjtz abcntz cdfjrz fhjntz bfhntz bdhjtz.

[00155] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 35 solutions for each combination includes the following combinations:

dghrst adirst.

[00156] The combination set may also include: cdhjrz abdfsz djmotz chjotz bdmntz adgitz acfitz dghitz abfntz dfjntz cfghtz bcdotz.

[00157] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 36 solutions for each combination includes the following combinations:

adjrtu bdhstu dgnrtu diostu.

[00158] The combination set may also include: afhntz achotz cfnotz.

[00159] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 37 solutions for each combination includes the following combinations:

dgotvw cdhstv adfrtv dintvw cdjstv bdfrtv.

[00160] The combination set may also include: cdfntz ghjntz abiotz cdfotz. [00161] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 38 solutions for each combination includes the following combinations:

bdituv cefrst dgjtuv bcfrtu cfjstu.

[00162] The combination set may also include: adijtz bdghtz.

[00163] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 39 solutions for each combination includes the following combinations:

cdhrst adfrst cfnstu cfortu.

[00164] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 40 solutions for each combination includes the following combinations:

cdfrst bdgtuv dijtuv.

[00165] The combination set may also include: abfotz chjntz.

[00166] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 42 solutions for each combination includes the following combination:

adhrst.

[00167] The combination set may also include: adhntz adhotz.

[00168] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 43 solutions for each combination includes the following combinations:

cdjstu bdfrtu.

[00169] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 44 solutions for each combination includes the following combinations:

defrtu cdestu.

[00170] The combination set may also include: bcfntz dehjtz cdghtz adeftz abdetz adfitz cdehtz cfjotz. [00171] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 47 solutions for each combination includes the following combinations: diotvw dgntvw.

[00172] The combination set may also include: aceftz acfjtz bcdfrz bdehtz adhitz adejtz adghtz cefhtz bcfhtz bcdjtz cdfjsz bdfjtz.

[00173] In an embodiment, an exemplary combination set for a 3x3x3 cube (e.g., shape 302) that corresponds to 59 solutions for each combination includes the following combinations:

abdtuv dhjtuv.

[00174] The combination set may also include: cdfhtz acdftz.

[00175] In an embodiment, an exemplary combination set for a 3x3x4 cube (e.g., shape 304) that corresponds to 2 solutions for each combination includes the following combinations:

bhkl-tv bglm-tu bgkm-tu bgio-tu bgil-uv bgik-uv ajkl-tv aelm-uw fklm-ru bfkl-tv fjkl-tw fjkl-tv aeil-tw bfgk-tw film-tv fikm-tw fikl-su fikl-ru fgjk-tu fgil-rt fgik-st cjkl-tv belm-tv bekl-vw bekl-uv bekl-sv klmo-ru klmo-rt.

[00176] In an embodiment, an exemplary combination set for a 3x3x4 cube (e.g., shape 304) that corresponds to 3 solutions for each combination includes the following combinations:

bglo-tu bglo-st bgkm-tv bgkm-st bgkl-vw bgkl-sv bgjk-tv bgio-tw aelm-tv aekl-st aekl-rt bfkl-tw aeil-rt fikl-uv fijk-tu fhkl-tu fgkm-ru fgkl-su fgjk-tw cjkl-tw bekm-rt bekl-rv klno-st klno-rt beik-tv aikl-uv begl-tv begl-st.

[00177] In an embodiment, an exemplary combination set for a 3x3x4 cube (e.g., shape 304) that corresponds to 4 solutions for each combination includes the following combinations: bhkl-st bhil-tw bhil-st bhik-tu bglm-tv bgkn-tw bgkl-rw bgjl-tv bgjl-st bgjl-rt bgjk-st bgim-st bgil-sv bgik-su bghl-tu bghk-tu ajkl-rt aeik-tu bfgk-tu fikm-tu fijl- tu cklm-su fgkm-tu fgkl-uv aegl-rt aegk-st bekm-tv klno-tv.

[00178] In an embodiment, an exemplary combination set for a 4x4x4 cube (e.g., shape 304) that corresponds to 2 solutions for each combination includes the following combinations:

-eij+s -eij+w -abi+s -acf+w -cde+s -ade+r -agi+s -cdh+r -dhj+w -cij+w -cio+w -cgo+w - ade+s -cdh+u -aci+s -cdi+t -ijo+r -cdi+u -cdj+v -ade+w -cdo+v -hjo+s -cdo+w -dij+s - adf+w -adg+w -cef+u -aci+w -aef+v -cem+w -aij+w -adi+w.

[00179] In an embodiment, an exemplary combination set for a 4x4x4 cube (e.g., shape 304) that corresponds to 3 solutions for each combination includes the following combinations:

-acf+v -adf+s -abi+w -adg+s -cdj+r -acg+w -cdo+r -adi+s -agi+u -adi+u -bce+s -fho+r - fho+w -abo+s -aci+v -bce+w -bcf+u -bch+s -abj+r -adj+s -afg+s -afi+r -cij+u -afi+s - adj+w -abc+u -chj+s -cho+w -afo+s -abh+v -abk+u -abf+t.

[00180] In an embodiment, an exemplary combination set for a 4x4x4 cube (e.g., shape 304) that corresponds to 4 solutions for each combination includes the following combinations:

-abm+u -fjo+v -aco+u -adf+r -cdj+w -cdm+v -dgi+s -acd+u -abm+v -cdo+u -fio+v - bem+w -abe+t -bjo+s -bjo+u -bjo+w -agi+w -agj+r -afg+u -acf+r -bde+w -bdh+w -bdo+s -adh+s -fho+s -bho+s -afi+t -adh+v -dfj+v -adh+w -afj+r -afj+w.

F. Exemplary Computing Systems

[00181] Fig. 15 is a block diagram of an exemplary computing system 1500 that may be used to perform the processes and operations described herein. The computing system 1500 may include one or more of each of bus 1502, processing device 1504, I/O 1506, memory 1508, storage device 1510, input device 1512, mouse 1514, camera 1516, keyboard 1518, touch sensor 1520, accelerometer 1522, output device 1524, display 1526, speaker 1528, and printer 1530.

[00182] The processing device 1504 may include a conventional processor or microprocessor that is configured to interpret and execute a set of instructions. The processing device 1504 may communicate with each of the other components in the computing device 1500 via the bus 1504, such to obtain instructions stored in the memory 1508 and/or the storage device 1510. The processing device 1504 may further be configured to receive inputs from the input device 1502, and to provide outputs via the output device 1524. The bus 1504 may permit communication between the components of the computing device 1500.

[00183] The memory 1508 may include RAM and/or ROM. The storage device

1510 may include magnetic hard drives, flash media, magnetic media, optical media, or another type of physical device that stores information for the processing device 1504. The storage device 1510 may include tangible machine-readable media and/or the corresponding drive for reading and/or writing to the machine-readable media. The memory 1508 and/or the storage device 1510 may store a set of instructions detailing a method that when executed by one or more processing devices cause the one or more processing devices to perform the method.

[00184] The input device 1512 may be used by a user to provide information to the processing device 1504. The input device 1512 may include one or more of the mouse 1514, camera 1516, keyboard 1518, touch sensor 1520, and/or accelerometer 1522. The output device 1524 may be used by the processing device 1504 to provide audio and/or visual output to one or more users. The output device 1524 may include the display 1526, speaker 1528, and printer 1530.

[00185] The I/O 1506 may be any device that permits the processing device 1504 to communicate with other devices and/or networks. For example, the I/O 1506 may include a modem, a network card, or other interface. The I/O 1506 may permit communication with wired, wireless, and/or optical systems (e.g., Bluetooth, USB, etc.).

The I/O 1506 may further permit peripheral devices to be connected to the computing device 1500, or to pair the computing device 1500 with other computing devices.

[00186] Fig. 16 includes a user computing device 1602, a network 1604, and a server 1606. The user computing device 1602 and/or the server 1606 may be computing devices 1500. The network 1604 may be a local area network, a mesh network, the internet, or other network.

[00187] Fig. 16 illustrates an example of a user computing device 1602 that may communicate with the server 1606 via the network 1604. The systems and methods described herein may operate entirely on the user computing device 1602 or the server 1606, or may involve operation on both.

[00188] While the foregoing written description of the invention enables one of ordinary skill to make and use what is considered presently to be the best mode thereof, those of ordinary skill will understand and appreciate the existence of variations, combinations, and equivalents of the specific embodiment, method, and examples herein. The invention should therefore not be limited by the above described embodiments, methods, and examples, but by all embodiments and methods within the scope and spirit of the invention.

[00189] In addition, although only some embodiments of the present invention have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the embodiments without materially departing from the novel teachings and advantages of this invention. Accordingly, all such modifications are intended to be included within scope of this invention.

Claims

CLAIMS WHAT IS CLAIMED IS:

1. A system to form at least one object based on calibrated degrees of difficulty, the system comprising:

a shape set comprising a first group and a second group, the first group including a plurality of first shapes, and the second group including a plurality of second shapes,

the shape set including a first combination set that forms a first object using first shapes and second shapes, each combination in the first combination set having a first quantity of solutions that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty.

2. The system of claim 1, the shape set comprising a third group that includes at least one third shape.

3. The system of claim 1, the shape set including a second combination set that forms the first object using first shapes and second shapes, each of the combinations in the second combination set having a second quantity of solutions that form the first object, the second quantity corresponding to a second degree of difficulty.

4. The system of claim 3, wherein the first quantity of solutions is lower than the second quantity of solutions, and the second degree of difficulty is lower than the first degree of difficulty.

5. The system of claim 3, wherein each of the first combination set and the second combination set includes at least two different solutions.

6. The system of claim 3, wherein each of the first combination set and the second combination set includes at least five different solutions.

7. The system of claim 1, the shape set including a third combination set that forms a second object using first shapes and second shapes, each of the combinations in the third combination set having a third quantity of solutions that form the second object, the third quantity corresponding to a third degree of difficulty.

8. The system of claim 7, wherein the first quantity of solutions is lower than the second quantity of solutions, and the second degree of difficulty is lower than the first degree of difficulty, and wherein the second quantity of solutions is lower than the third quantity of solutions, and the third degree of difficulty is lower than the second degree of difficulty.

9. The system of claim 1, wherein each first shape has a first volume, and each second shape has a second volume.

10. The system of claim 1, wherein each first shape is equivalent to a first structure composed of five unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the first structure.

11. The system of claim 10, wherein each second shape is equivalent to a second structure composed of four unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the second structure.

12. The system of claim 1, wherein the each solution of the first combination set uses three first shapes and three second shapes.

13. The system of claim 1, wherein each of the first shapes is nonplanar.

14. A method of forming at least one object based on calibrated degrees of difficulty using a shape set that includes a first group and a second group, the first group including a plurality of first shapes, and the second group including a plurality of second shapes, the shape set including a first combination set that forms a first object using first shapes and second shapes, each combination in the first combination set having a first quantity of solutions that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty, the method comprising:

providing the first combination set to a player, the first combination set including a plurality of combinations that form the first object using first shapes and second shapes, each of the combinations within the first combination set having the first quantity of solutions that form the first object, the first quantity being greater than one;

providing the first shapes that are used in the combinations of the first combination set; and

providing the second shapes that are used in the combinations of the second combination set.

15. The method of claim 13, the shape set including a second combination set that forms the first object using first shapes and second shapes, each of the solutions in the second combination set having a second quantity of solutions that form the first object, the second quantity corresponding to a second degree of difficulty.

16. The method of claim 13, wherein the first quantity of solutions is lower than the second quantity of solutions, and the second degree of difficulty is lower than the first degree of difficulty.

17. The method of claim 13, wherein each of the first combination set and the second combination set includes at least two different solutions.

18. The system of claim 1, wherein each first shape is equivalent to a first structure composed of five unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the first structure.

19. The system of claim 10, wherein each second shape is equivalent to a second structure composed of four unit cubes that each have at least one full face matched against a corresponding full face of another unit cube of the second structure.

20. A computer based method of forming at least one object based on calibrated degrees of difficulty using a shape set that includes a first group and a second group, the first group including a plurality of first shapes, and the second group including a plurality of second shapes, the shape set including a first combination set that forms a first object using first shapes and second shapes, each solution in the first combination set having a first quantity of solutions that form the first object, the first quantity being greater than one and corresponding to a first degree of difficulty, the method comprising:

using a processing device and a display, providing the first combination set to a player, the first combination set including a plurality of combinations that form the first object using first shapes and second shapes, each of the combinations within the first combination set having the first quantity of solutions that form the first object, the first quantity being greater than one;

providing the first shapes that are used in the combinations of the first combination set; and

providing the second shapes that are used in the combinations of the second combination set.