US3667760A - Puzzle forming regular geometric figures from a changeable number of pieces having sequentially increasing surface areas - Google Patents

Abstract

A PUZZLE HAVING A SET OF PIECES, SAID PIECES HAVING SEQUENTIAL UNIT SURFACES AREA DIFFERENCES AND BEING COMBINABLE TO FORM A COMPOSITE RECTANGULAR FIGURE. THE RECTANGULAR FIGURE THUS FORMED HAS A SURFACE AREA WHICH IS ONE-HALF THE TOTAL NUMBER OF PIECES TIMES THE SUM OF THE BEGINNING AND ENDING UNIT AREAS OF THE PIECES. AN ADJUSTABLE FRAME IS PROVIDED TO ACCOMMODATE LARGER AND SMALLER NUMBERED SETS OF PIECES FORMING CORRESPONDING LARGER AND SMALLER RECTANGULAR FIGURES. ALTERNATIVE PIECES HAVING THE SAME AREA BUT OF DIFFERENT SHAPE THAN INDIVIDUAL ONE OF THE PIECES MAY BE PROVIDED. RESPECTIVE PIECES MAY BE OF FOUR DIFFERENT COLORS SO SELECTED THAT WHEN THE PIECES FORM SAID RECTANGULAR FIGURE NO PIECES OF THE SAME COLOR ARE ADJACENT EACH OTHER.

Description

June 6, 1972 w. NELSON 3,667,760

I'U'L'ALFJ FORMING REGULAR GEOMETRIC FIGURES FROM A CHANGEABLE Filed Jan. 26, 1970 FIG. IA

NUMBER OF PIECES HAVING SEQUENTIALLY INCREASING SURFACE AREAS 3 Sheets-Sheet 1 mo "mzfWQ N. g

lNl/E/VTOR WINSTON L. NELSON June 6, 1972 w. NELSON 3,667,760

PUZZLE FORMING REGULAR GEOMETRIC FIGURES FROM A CHANGEABLE NUMBER OF PIECES HAVING SEQUENTIALLY INCREASING SURFACE AREAS Filed Jan. 215, 1970 3 Sheets-Sheet :3

FIG. /5

F/G. 2A FIG. 2B

June 6, 1972 w. L. NELSON 3,667,760

PUZZLE FORMING REGULAR GEOMETRIC FIGURES FROM A CHANGEABLE NUMBER OF PIECES HAVING SEQUENTIALLY INCREASING SURFACE AREAS Filed Jan. 23, 1970 5 Sheets-Sheet FIG. 3A

United States Patent PUZZLE FORMING REGULAR GEOMETRIC FIGURES FROM A CHANGEABLE NUMBER OF PIECES HAVING SEQUENTIALLY IN- CREASING SURFACE AREAS Winston L. Nelson, 24 Erskine Drive, Morristown, NJ. 07960 Filed Jan. 23, 1970, Ser. No. 5,331 Int. Cl. A63f 9/10 US. Cl. 273-157 R 10 Claims ABSTRACT OF THE DISCLOSURE A puzzle having a set of pieces, said pieces having sequential unit surfaces area diiferences and being com binable to form a composite rectangular figure. The rectangular figure thus formed has a surface area which is one-half the total number of pieces times the sum of, the beginning and ending unit areas of the pieces. An adjustable frame is provided to accommodate larger and smaller numbered sets of pieces forming corresponding larger and smaller rectangular figures. Alternative pieces having the same area but of different shape than individual ones of the pieces may be provided. Respective pieces may be of four difierent colors so selected that when the pieces form said rectangular figure no pieces of the same color are adjacent each other.

BACKGROUND OF THE INVENTION This invention relates to puzzles and, more particularly, to the type of puzzle which is solved by arranging playing pieces within a geometric frame of reference.

In the usual puzzle that is played within a geometric frame of reference, the desired solution is a unique arrangement of the playing pieces. Such a puzzle is of limited flexibility. The solution is generally realized through a trial and error arrangement of the pieces, affording a player little opportunity to employ a reasoned procedure for placing the various pieces. Once the puzzle is solved, its subsequent appeal is greatly diminished.

Moreover, the level of complexity of the ordinary geometric puzzle is beyond the control of the players. Where the puzzle is too ditficult it remains incomplete. Where the puzzle is too simple, it will not have the desired player appeal.

Accordingly, it is an object of the invention to provide a flexible puzzle with playing pieces that are arranged to form a pattern within a specific geometric frame of reference. Another object of the invention is to achieve a geometric puzzle which, for a specific configuration, has a variety of solutions.

Still another object of the invention is to provide a geometric puzzle with a recognizable interrelation among the playing pieces, so that the solution is not completely a matter of trial and error.

A further object of the invention is to provide a geometric puzzle which can be tailored in difliculty to the ability of the player. A related object is to provide a puzzle with gradations of difliculty in accordance with the number of pieces employed.

SUMMARY OF THE INVENTION In accomplishing the foregoing and related objects, the invention provides a puzzle in which a plurality of playing pieces is provided, each representing a diflerent surface area. At least one of the pieces is of a first kind, for example, a one-dimensional array of one or more unit areas, and forms a portion of a prescribed geometric 3,667,760 Patented June 6, 1972 figure. The remaining pieces are of a second kind, for example, two-dimensional arrays of unit areas that form the remainder of the puzzle.

For one embodiment of the invention, the pieces of the first kind have playing surfaces measuring lxp square units, where p is a diiferent integer for each piece, such as a prime number; and the pieces of the second kind have playing surfaces measuring m n=q square units, where m and n are integers greater than unity and q is a different integer for each piece, for example, a non-prime number.

In accordance with one aspect of the invention, the areas represented by the various playing pieces are in a number sequence which may be a set of odd and even integers, a set of even integers alone, or a set of odd integers alone. The number of integers, and hence the number of pieces in the puzzle, determines the complexity and size of the resulting geometric figure. The size of the figure is changed by altering the sequence. Where the sequence is expanded, there is a corresponding enlargement of the puzzle. Where the sequence is contracted, there is a corresponding reduction in the puzzle.

In general, the number of units of area in the overall figure is one-half of the number of pieces times the sum of the areas represented by the first and last pieces. This relationship can be used as a guide in forming the outline of a desired quadrilateral figure. Hence, where the puzzle has twenty-five pieces ranging in area from one square unit to twenty-five square units, one possible geometric framework measures twenty-five units by thirteen units.

In accordance with another aspect of the invention, a frame can be provided in order to establish an outline for the puzzle. Since the size of the puzzle depends upon the number of pieces, the frame is desirably made adjustable.

In accordance with still another aspect of the invention the pieces of the puzzle are selectively colored with one of four colors in such a way that one arrangement of the pieces will have no adjoining pieces of the same color.

In accordance with a further aspect of the invention, alternative pieces, for a puzzle with a specified number of pieces, are provided having the same playing surface areas as the original pieces, but different shapes, to permit still other solutions of the puzzle.

BRIEF DESCRIPTION OF THE DRAWINGS Other aspects of the invention will become apparent after considering several illustrative embodiments taken in conjunction with the drawings in which:

FIG. 1A is a perspective view of a puzzle in accordance with the invention;

FIG. 1B is a plan view of an alternative assemblage of the playing pieces of FIG. 1A;

FIG. 2A is a plan view of an alternative assemblage of pieces corresponding to even integers in the puzzle of FIG. 1A;

FIG. 2B is a plan view of a complementary portion of the puzzle of FIG. 2A formed by pieces representing odd integers only; and

FIGS. 3A through 3D are plan views of subordinate puzzles of increasing degrees of difliculty formed from various sequences of pieces in the puzzle of FIG. 1A, with FIG. 3A showing a subordinate puzzle formed by the first eight pieces of FIG. 1A,

FIG. 3B showing a subordinate puule formed by the first eleven pieces of FIG. 1A,

FIG. 3C showing a subordinate puzzle formed by the first thirteen pieces of FIG. 1A, and

FIG. 3D showing a subordinate puzzle formed by the first fifteen pieces of FIG. 1A.

, pasoarrrrou on ,THE ILLUSTRATIVE EMBODIMENTS Turning to the drawings, FIG. 1A shows a puzzle -1 in accordance with the invention constituted by'various playing pieces in the course of being assembled within an adjustable frame to form a rectangular quadrilateral.

The puzzle 30-1 illustratively has twenty-dive pieces which are sequentially numbered from 1 through 25. Each piece has a playing surface delineated with a number of square units corresponding to its numerical designation. Thus the representative playing pieces 40-25, 40-22 and 40-20 respectively have playing surfaces with twenty-five, twenty-two and twenty square units. When all of the playing pieces 40 have been assembled with the frame 50 adjusted-as shown in FIG. 1A, they form a rectangular figure that is twenty-five units by thirteen units.

In addition, the various playing pieces 40 are of two types. Pieces of the first type, such as the thirteen unit piece 40-13, have unit areas linearly arrayed. The remaining pieces, of the second type, such as the twenty-one unit piece 40-21, have their unit areas arrayed in two dimensions. The particular piece 40-21 measures seven units in length and three units in Width.

. It is the correspondence between the numerical designations .of the playing pieces and the square units of area of their. playing surfaces that permits the pieces to be arranged. in a desired geometric figure on other than a trial and-error basis. Further, it is the use of two different types of pieces that gives the puzzle the desired degree of challenge.

The general scheme for the two types of pieces employed in the puzzle 30-1 is that those corresponding to prime numbers are of the first type, i.e. are elongated with their unit areas disposed in a linear array; while the remaining pieces (corresponding to non-prime numbers) are of the second type, i.e. with unit areas arrayed in two dimensions. A prime number is one having only itself or unity as a factor, e.g. 1, 2, 3, 5, etc.

In mathematical terms, the playing pieces of the first type have surface areas measuring l p square units, where p is a prime number. The remaining pieces have surface areas measuring q=m n square units, where q is a non-prime number formed by non-unity factors In and n.

Accordingly, in the twenty-five piece puzzle of FIG. 1A, the pieces of the first type are associated with the prime numbers 1, 2, 3, 5, 7, 11, 13, 17, 19'and 23', while the pieces of the second type are associated with the nonprime numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 and 25.-

Where the non-prime numbers are factorable into more than two prime factors, the factors are advantageously selected to make the pieces as square as possible, i.e. as close to each other as possible on the number scale. For example,the piece 40-20 with a playing surface of twenty square units, factorable into prime factors five, two and two, has a surface measuring five units by four units instead of the alternative ten units by two units. j The puzzle 30-1 of FIG. 1A is shown in the final stages of completion, with the pieces 40-13 and 40-21 remaining to be inserted. As noted previously, the complete array 30-1 measures twenty-five units by thirteen units. These dimensions are obtainable from Equation 1 below, which gives the numerical value for the sum S of n integers.

where n is the number of integers, a is the first integer of the sequence, and b is the last integer of the sequence.

It will be appreciated that the rectangular quadrilateral shown in FIG. 1A is only one of the possible geometric figures for the puzzle 30-1. Alternatively, for example, the geometric framework may take the form of a rectangular quadrilateral which measures sixty-five units by five units.

In order to accommodate quadrilaterals having other sizes, the frame 50 of FIG. 1A is adjustable. For that purpose, there are four frame members 50-1 through 50-4 with spaced apertures 52 and separable fasteners 51. The size of the frame is readily changed by separating the fasteners 51 and rearrangeing the members 50-1 through 50-4 as'desired. The particular frame 50 shown in FIG. 1A is representative only, and a suitable frame may be realized in a wide variety of other ways.

Besides permitting other overall geometric figures, the pieces 40 of the puzzle 30-1 in FIG. 1A may be rearranged to provide other solutions within the original geometric framework. One such alternative is provided in FIG. 1B by the version 30-2 of the puzzle which uses the reverse sides of the pieces 40 in FIG. 1A.

The puzzle 30-2 omits the frame 50 of FIG. 1A and the player instead can rely upon the markings of the various pieces in realizing the desired twenty-five unit by thirteen unit rectangular framework.

In addition, the pieces 40 of FIG. 1B have their playing surfaces variously colored. This illustrates a further aspect of the invention by which the constituent quadrilateral pieces can be assembled so that no two adjoining pieces have the same color. To that end, four colors, illustratively red, yellow, green and blue, are selectively applied to the pieces in the manner as summarized by Table I.

TABLE I Color: Piece values lRed 5, 7, 9, 1'7, 20, 24 Yellow 2, 4, 12, 13, 16, 22 Green 3, 11, 15, 18, 19, 25 Blue 1, 6, 8, 10, 14, 21, 23

in order to realize the desired rectangular configuration.

Another form of quadrilateral arrangement in accordance with the invention is shown by the twenty-five piece puzzle 31 of FIGS. 2A and 2B, which is shown partitioned into an even section 31-1 (FIG. 2A) and an odd section 31-2 (FIG. 2B).

The even section 31-1 is formed by the even valued pieces 40 of FIG. 1A. From Equation 1 above, it is apparent that the even valued pieces can form a thirteen unit by twelve unit quadrilateral, since It is 12, a is 2 and b is 24. It is to be noted that the even section 31-1 of the puzzle 31 includes a single prime number 2, the remainder being non-prime. Consequently the section 31-1 includes a single piece which is a two-unit linear array and eleven pieces which are two-dimensional arrays of unit areas.

The odd section 31-2 of the puzzle 31 is formed from odd numbered pieces. The pieces representing the prime number 1 and the non-prime numbers 9, 15, 21 and 25 are taken from the puzzle 30-1 of FIG. 1A. The pieces representing the remaining prime numbers 3, 5, 7, ll, 13, 17, 19 and 23 are of special notched construction. In any event there are two types of pieces: A piece of the first type representing the number 1 as a linear array of a single unit and the remaining pieces of the second type with surface areas that are two-dimensional arrays of unit areas corresponding to both prime and non-prime numbers.

The flexibility of the puzzle 30-1 of FIG. 1A is illustrated by subordinate version 30-3 through 30-6 shown in FIGS. 3A through 3D.

The version 30-3 of FIG. 3A is a rectangular figure constructed using the first eight pieces of FIG. 1A. From Equation 1, above, it is apparent that these pieces can form a rectangle which measures nine units by four units. Alternative rectangular constructs (not shown) measure eighteen units by two units and twelve units by three units. The version 30-3 may be employed, for example, by a beginning player who finds the complexity of a larger scale puzzle too great. Beginning with the version 30-3 successive pieces can be added to the puzzle to realize succeedingly more diflicult versions as the skill and understanding of the player increases.

Thus, a further stage of the puzzle is the version 30-4 of FIG. 3B in which the version 30-3 of FIG. 3A has been supplemented by three playing pieces 40-9, 40-10 and 40-11. Accordingly, there are eleven pieces in the version 30-4 which according to Equation 1 above, can form a rectangular figure that measures eleven units by six units. It is to be understood that the version 30-3 of FIG. 3A can be supplemented by one additional piece at a time.

To achieve a still higher degree of complexity, three additional pieces 40-12, 40-13 and 40-14 in FIG. 3C result in another version 30-5 of the puzzle. As with the transition from FIGS. 3A to 3B, the transition from FIGS. 3B to 3C requires a rearrangement of the playing pieces.

A final illustrated version of still greater complexity is formed in FIG. 3D by adding a fifteenth piece 40-15 to achieve a version 30-6.

It is apparent that the reverse procedure may be followed by which succeeding versions of the desired rectangular quadrilateral are formed by the removal of pieces and the progression of play ranges, for example, from the version 30-6 of FIG. 3D to the version 30-3 of FIG. 3A.

The foregoing detailed description is illustrative only and various changes in parts, shapes and proportions, and the substitution of equivalents for the elements and constituents shown and described, can be made by those of ordinary skill in the art without departing from the spirit and scope of the invention.

What is claimed is:

1. A puzzle comprising a set of at least nine playing pieces which are combinable to form a rectangular array measuring at least three units by fifteen units or at least nine units by five units, each piece having a different playing surface area in square units equivalent to a different cardinal member,

said pieces forming a consecutive non-duplicative set extending from a beginnnig number to an ending number,

and a frame for said playing pieces defining the outline of said rectangular array having a surface area which is one-half of the total number of pieces times the sum of the beginning and ending numbers.

2. A puzzle as defined in claim 1 wherein each of said playing pieces is selectively colored with one of four colors,

and said pieces are combinable within said frame to form said rectangular array in which no two adjacent pieces have the same color.

3. A puzzle as defined in claim 1 wherein a further, alternative playing piece is provided having the same surface area as one of said pieces but a different shape.

4. A puzzle as defined in claim 1 wherein a further playing piece is included having a playing surface area in square units equivalent to said ending number plus unity,

and said frame is adjustable to define a playing surface area which is one-half of the total number of pieces times the sum of the beginning and ending numbers plus unity.

5. A puzzle as defined in claim 1 wherein at least one of said playing pieces has a playing surface area measm'ing l p square units, where p is a different prime number for each such piece,

and the remainder of said pieces have playing surface areas measuring m n=q square units, where m and g n are integers greater than unity and q is a difiernet I non-prime number for each such piece.

6. A puzzle comprising a set of at least nine playing pieces,

each piece having a different playing surface area in square units equivalent to a different cardinal number,

said pieces forming a consecutive non-duplicative set extending from a beginning number to an ending number and being combined to form a rectangular array measuring at least three units by fifteen units or at least nine units by five units, said array having a surface area which is one-half of the total number of pieces times the sum of the beginning and ending numbers.

7. A puzzle as defined in claim 6 wherein said playing pieces are selectively colored with one of four different colors and are combined to form said prescribed rectangular array in which no pieces of the same color are adjacent each other.

8. A puzzle as defined in claim 6 wherein a further alternative piece is provided having the same surface area as one of said pieces but a different shape.

9. A puzzle as defined in claim 6 wherein a further playing piece is included having a playing surface area in square units equivalent to said ending cardinal number plus unity,

and all of the pieces are combined to form a rectangular array having a surface area which is one-half of the number of pieces times the sum of the beginning and ending numbers plus unity. 10. A puzzle as defined in claim 6 wherein at least one of said playing pieces has a playing surface area measuring lXp square units, where p is a different prime number for each such piece, and the remainder of said pieces have playing surface areas measuring m n=q square units, where m and n are integers greater than unity and q is a different non-prime number for each such piece.

References Cited UNITED STATES PATENTS 3,131,488 5/1968 Slater 35-31 D 3,414,986 12/1968 Stassen 35-31 D 2,003,072 5/1935 Eynon 273-157 R 2,680,306 6/1954 Moyer 35-70 X FOREIGN PATENTS 169,653 12/1951 Austria. 1,090,413 11/ 1967 Great Britain.

ANTON O. OECHSLE, Primary Examiner US. Cl. X.R.

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