Mathematically formulated and androgynously linked polygonal and polyhedral gamepieces



United States Patent [72] Inventors [54] MATHEMATICALLY FORMULATED AND ANDROGYNOUSLY LINKED POLYGONAL AND POLYHEDRAL GAMEPIECES 7 Claims, 21 Drawing Figs. [52] 11.8. C1. 273/137, 273/131, 273/136, 273/157 [51] Int. Cl. A63t 3/00 [50] Field ofSearch 273/137, (Cursory), 156, 157(Cursory), 153, 160; 46/16, 24 [56] References Cited UNITED STATES PATENTS 139,928 6/1873 Stevens 273/137 331,652 12/1885 Richards 273/137 407,374 7/1889 Smith 273/153X 588,705 8/1897 Nelson 273/ 160 176,532 4/1876 Hughes 273/157X 284,037 8/1883 McC1eary.. 273/156 1,558,165 10/1925 l-laswell 273/137 2,072,605 3/1937 Palmer 273/137 2,202,592 5/1940 Maxson 2,937,874 5/ 1960 Ellison 273/137 ABSTRACT: A plurality of polygonal bodies forming game pieces are positioned with peripheral sides contiguous and identically matching in size and shape. The center portion of each body has a member of the same set-group of indicia formed thereon with the body center portion indicia having a predefined association one to another between adjacent of these bodies. In one form of the invention, each peripheral side of each body also has a member of the same set-group of indicia formed thereon with these peripheral side indicia being the same indicia between the contiguous sides of adjacent bodies, and with there being no relationship between the peripheral side indicia of the various bodies and the center portion indicia of those bodies. In another form of the invention, rather than requiring the body peripheral side indicia, the peripheral of contiguous bodies are interfitting, preferably intcrlocking, and identical as between contiguous peripheral sides of adjacent bodies. PATENTEDUEBI 5.9m 5 1 sum 2 or 7 Era 2. lAn zurozs 14305597 A. MV/LL/A/ms, L/A 1%55 E. BLA l/Q MAHOA/EY, HALBEET & HOEA/BA/(EX? ATTORNEYS PATENTED am 51970 SHEET 3 BF 7 /22 fir /30 HHIIHHIIH HHIIIHIIIIII II HINT a a Z W Z 3 /A L E L WW Z 5 1m T o 2 3% RJM 4 w 2 w Avroemsys m'rusmncsttvromutsrsmun ANDROGYNOUSLY unxsn POLYGONAL AND POLYHEDRAL GAMEPIECES generally consisting of enantiomorphous pairs of arrange- 1 ments that are so permissive of assembly into various twodimensional bodies that said bodies become the physical summations of the strategies employed in resolving the conflicts of interest of game opponents, since the said game pieces also are without memorizable peculiarities and statistical bias, and are partially incompatible in an unpredictable manner as a result of the choice and chance used in determining positions in two-dimensional assembly. According to the present invention there are provided related sets of related game pieces wherein the segments that outline each of the game pieces in portions of any formulated set constitute distinguishable mirrorimage, or enantiomorphous, arrangements of the segments that outline the game pieces in the remaining portions of the set, and each game piece is distinguishable from every other game piece. The bounding segments, or game piece sides, are distinguishably ordered according to the formulated arrangements of this invention so as to make possible the assembly of twodimensional surfaces in many different ways, while also introducing unpredictable incompatibilities to stalemate game tactics and strategies. Thus, the assembly of the game pieces constitutes a puzzle, a game, and a gameboard of interlocked game pieces that affects the pursuit of any game played thereon. Since the assembly of the game pieces depends upon the exercise of both choice and chance, with the set of game pieces providing unbiased and unpredictable opposition to two-dimensional assembly, the object of this invention is to provide competitively assemblable two-dimensional playing surfaces constituted of pluralities of game pieces whose adjacentbounding sides can be strategicia'llymated commensurate with the abilities of the players of the games. In contrast, those prior art games which are constituted of a plurality of game pieces are incapable of providing such results. For example, certain such games of the prior art, that utilize multisided game pieces in which theiside portions are provided with indicia and which resemble the game pieces of the present invention, have the important and critical difference that incompatibiliies between such game pieces depend on the differences inherent in combinations. Such ordinary mathematical combinations of indicia undesirably affect the assembly of game pieces into multidimensional bodies since the symbol differences that are definitive of combinations are favorable only to the linear, or one-dimensional, as- sembly exemplified by dominoes. Accordingly, it is seen that the present invention is primarily concerned with the novel relationships of the uniquely formulated arrangements, or distinguishably permuted mirror-image arrangements, of the game piece sides that are peculiarly suited to the assembly of twoand three-dimensional bodies. Also in contrast, the indicia combinations of prior art games loclt their game pieces into fixed relation ship in two-dimensional assembly almost as surely as do the individualistic curves, lengths, tongues and grooves of jig-saw punles. In fact, the jig-saw puzzle pieces that superficially resemble the game pieces of this invention in two-dimensional assembly are incapable of assembly in more than one way, and therefore, the assembly of such a puzzle for a gameboard serves only to delay the start of any game that might beplayed on its unalmay be made interchangeably interlockable, but this constitutes one of the related objects of this invention; or said pieces may be altered to be selectivelyinterchangeable, but this constitutes another related object of this invention; or, said pieces may be formed so that androgynous sides may be mated to their own identity or identical sides may be mated by means of vincula, but these are still other related objects of this invention. it is the mechanics of these discoveries coupled with the physical relationships of the uniquely formulated arrangements of the bounding edges of the game pieces that produce the unique games, puzzles, answer patterns, and gameboards of this invention. I. One of the more general objects of this invention is to provided new and novel game pieces suitable for a variety of new games. Each of the game pieces, for example, may have a generally square shape outlined by four distinguishably different sides arranged in the formulated manner of this invention. Game pieces, without memorizable peculiarities, that are permissive of two-dimensional assembly while being statistically unbiased and upredictably incompatible, are essential to the new games of skill, of which the following are exemplary: l. In one application, the portions of the faces of the game pieces surrounded by the distinguishable sides collectively constitute colored squares for forming a bicolored checkered pattern, and the object is to assemble the game pieces in the form of a gameboard for playing a checkertype game, with each opponent attempting to construct the checkerboard to be most advantageous to his own subsequent checker play. For instance, one of the obvious questions a player might investigate is whether it is wise to have only one square in his king" row, thus reducing his own number of checkers by 25 percent while reducing his opponents chances for gaining king" by at least percent, predicated, of course, on his ability and luck in-so arranging the gameboard. 2. In another application, each distinguishably bounded square may contain a letter of the alphabet so that, by playing one game piece at a time, words of any length may be spelled in various directions through the mated sides of the adjoining pieces, the players building on their own or on anothers word, or portion of such word, thus eliminating the gameboard required to modify or inhibit spelling as in prior art spelling games. 3. Yet another application of these game pieces provides each of the uniquely bounded centers of the game pieces with an intelligibility relating, say, to the geography of the United States, as for example, the name of a city, or a Na tional Park, or a river, or a highway number, or to an operating instruction for the game, such as "lose one turn," move two squares in the direction of the arrow," etc. A gameboard is assembled by mating the distinguishable sides of game pieces while conforming to the relationships of the items of geography thereon. Of course, each player will attempt to, place the operating instructions" to favor his own pursuit of the game when the gameboard is finally assembled, and the surface can then be used in the conventional manner to play the game. As will be apparent, there are many other applications for the game pieces of this invention to suit different games for various levels of intelligence or mood,'and their use is limited only by the nature of the game intelligibilities applied to the exposed faces of the game pieces by those skilled in the game art. Another object of this invention is to provide the said game pieces with the compatibility for two-dimensional assembly required by games of which those just described are illustrative, but not limiting. It is obvious that the indicia utilized on the playing pieces of prior art games are pattern after dominoes, and have the peculiarities inherent in combinations, so that they are incapable of two-dimensional assembly except as easily memorized puzzles having few solutions; and terable assembled surface. Of course, the jig-saw puzzle pieces 75 players are unable to duplicate the many accomplishments of the game pieces of the present invention. The several following objectives are predicated on this new two-dimensional compatibility provided by objects unique enantiomorphous arrangements of the bounding sides of the polygonal game pieces of the present invention One of these objects is to provide game pieces that generally are without memorizable peculiarities. Generally, this is accomplished by minimizing the repetition of side indicia on any game piece, and by making every game piece compatible in some degree with every other game piece in the set. It is desirable, but not generally possible, that every game piece should be without repetitions and the definitive differences of the mathematical combinations used on prior art playing pieces, since both repetitions and combinations are memorizable peculiarities. Another of the said objects is to provide game pieces that are unpredictably incompatible in two-dimensional assembly. .T he distinguishably permuted arrangements of the bounding sides of the polygonal game pieces of the present invention which provide two-dimensional compatibility also provide the unpredictable incompatibilities. All of the game piece positions in two-dimensional array generally require that each game piece be consentaneous on more than one side, and therein lies the source of the unpredictable incompatibilities, which source is related mathematically to the rate of use, r, of sides to outline each game piece and the number of different sides, n, available for such use. It is to be noted that the angles between the segments that outline the game pieces can be angularly distorted without seriously disturbing the permuted relationships. The three prior objectives have a corollary object wherein are provided game pieces that are without statistical bias. It will be seen at once that game pieces designed with the twodimensional compatibility of this invention, which are also unpredictably incompatible in two-dimensional assembly and lack memorizable peculiarities, are capable of being statisti- Cally b ased-I ia. of out st s convenient y of Saying that games utilizing the game pieces of this invention will not favor one player over another in a long series of played games, and their use, therefore, results in new and novel games of skill. Consistent with the foregoing, it is yet another object of this invention to provide pairs of distinguishable enantiomorphous arrangements of the different sides of the game pieces conforming to the general formula: ABC 1u An C'B wherein is indicated in a specific order on one side of the operator -+the different sides of a game piece, while on the opposite side of the operator is indicated the distinguishable, enantiomorphous or mirror-image arrangement of the said sides to complete the pair. As will be seen, all the possible distinguishable enantiomorphous arrangements of the different sides of a set of game pieces constitute a definite quantity permuted in conformance to the said formula, wherein the quantity depends on the number of different sides n selected to outline the game pieces, and the rate r at which the sides are used to outline each game piece. Accordingly, it is an object of this invention to provide game pieces in sets conforming to the aforesaid forinula, with the total number of members of each set being determined by the formula (n-r) !r when n is equal to or greater than r for all distinguishably different ABC..... n An CB for any polygon. The usual factorial notation is employed wherein indicates the continued product of such designated number successively reduced by one unit to unity. Thus, if n 3 then n would be equal to 3X 2 X 1 and when n=rthe quantity l; and when n l the quantity 1 I, all according to mathematical convention for Permutations" as shown in the Encyclopedia Americanana as a readily available reference Another object of this invention is to provide game pieces, conforming to the aforementioned fonnulas, that are repetitively assemblable in two-dimensional array, so that pluralities of sets may be used to suit the particular requirements of any of a variety of games. The prior art never recognized the use of pluralities of its sets of combinations, but only numerical extensions or enlargements, as in the larger sets of dominoes. Such numerical extensions perpetuated the use of combinations, and made multisided" dominoes of prior art playing pieces, so the characteristics that make the game pieces of this invention uniquely applicable to many games remained undiscovered. Somewhat in the same vein, another object of this invention to provide game pieces conforming to the aforementioned formulas that are assemblable in two-dimensional array so that different kinds of sets, or segments of sets, may be used together to suit the particular requirements of any of a variety of games. As will be shown, the geometry of mixed shapes for game pieces, or segments of sets, is consistent with the algebra of arrangement. Compatible with the other objectives and characteristics of this invention, it is yet another object to provide game pieces with interlocks suitable for joining the said game pieces together for use in puzzles, games, and answer patterns, and for assemblable gameboards, that are a part of the games played on their assembled surfaces. The players of a game competitively assemble a gameboard comprised of the interlocking game pieces of this invention and then proceed to play a game, obviously related to that particular assemblage of the gameboard, on the just-assembled board. This is further described hereafter in the specification. Obviously, the game pieces must be assemblable in various unpredictable ways for such assembly to have meaning. This distinguishes the invention over the interlock of the jig-saw puzzle, since characteristically there is only one possible assembly for a jig-saw puzzle. The unique interlock of this invention is such that any game piece edge is androgynous or identical to the edge with which it interlocks, and also is such that the form of the interlocks on the several edges of each game piece can be made to conform to the formulated arrangements of this invention. Another object of this invention is to provided game pieces conforming to the aforementioned formulas that are definitive of the solids that resemble crystal structure, including the five classic solids. It is apparent that this requires that the said game pieces be physically adaptable to three-dimensional assembly, as will be shown, and that said game pieces must present distinguishable polygonal faces that are characteristic of the particular solid represented. Irregular shapes formed by sides of different lengths create larger sets wherein the differences of length may be treated as additional n in the formulas, depending on the mathematical logic involved. Again, angular distortion is compatible in twoand three-dimensional assembly with the permuted relationships, but in the five regular solids the polygonal game pieces must be geometrically regular polygons. Of course, an infinite variety of mosaics may be assembled with mixtures of the game pieces. Another object of the invention is to provide inexpensive and easily manufactured game pieces having compatible and distinguishable arrangements of their bounding sides such that compatibility in assembly of the game pieces that constitute a game is sufficient to evoke the interest of the average person, and yet is sufficiently limited to challenge the skill and ingenuity of the player with a superior intellect. Other objects and features of the present invention will become apparent to those skilled in the art from the following specification and the accompanying drawings wherein are illustrated various forms of the invention, and in which: FIG. 1 is a diagrammatic showing of a plurality of triangular game pieces utilizing enantiomorphous arrangements of three distinguishably different bounding sides showing multidimensional compatibility and incompatibility in its simplest characteristic form; FIG. 2 is a diagrammatic showing of a plurality of triangular game pieces utilizing enantiomorphous arrangements of five distinguishably different bounding sides used three at a time, and showing a formulated set arranged to illustrate repetitive assemblability; FIG. 3 is a diagrammatic showing of a plurality of square game pieces utilizing enantiomorphous arrangements of four distingu'ishably different bounding sides also arranged to illustrate repetitive assemblability; FIG. 4- is a diagrammatic showing of a plurality of pentagonal game pieces utilizing enantiomorphous arrangements of five distinguishably different bounding sides arranged to illustrate the repetitive assemblability of a game set and the segment of a set, the body formed by the four central pieces being also repetitively, assemblable and contiguous sides being shown .as superposed so that only one line of juncture is evident; FIG. 5 is a diagrammatic showing of a plurality of hexagonal pieces utilizing enantiomorphous arrangements of six different peripheral sides to illustrate the regular assemblability of an hexagonal subset; i FIG. 6 is a diagrammatic showing of a partial application of an hexagonal subset to a table game constituted of deformed playing cards; 1 FIG. 7 is a diagrammatic showing of characteristic forms of androgynous mating and interlocks; FIG. 8 is an isometric view of a cube showing the androgynous relationships of the faces of a cubic shell with the definitive application of a formulated set of the distinguishable enantiomorphous arrangements of this invention to the shell of one of the six regular solids; FIG. 9 is a developed view of the six faces of the cube illustrated in FIG. 8; f FIG. 10 is an isometric view of one side of a face piece of a shell of an assemblable cube; . FIG. 11 is an isometric view of the opposite side of the face piece illustrated in FIG. 10; ' FIG. 12 is an isometric view of four face pieces of the o'fFlG. 10 assembled to form an open-ended'cub'e; FIG. 13 is an exploded isometric view of a pair of face pieces disposed so as to nest together in interlocking relationship and, when so nested, to mate with other like face pieces to form a cube; FIG. 14 is an isometric view illustrating mating positions of the pairs of nested face pieces like that of FIG. 13; FIG. 15 is a fragmentary cross-sectional view taken on the line 15-15 of FIG. 14; FIG. 16 is a perspective view of an alternate arrangement of the projections and indentations of the face pieces of FIG. 13; FIG. 17 is a diagrammatic view of a partial octagonal subset illustrative of a table game somewhat resembling a playing card game but with the addition of peripheral indicia as a game factor; 1 FIG. 18 is a diagrammatic view of the two faces of the same game piece showing that different pictures applied to the two faces of a jig-saw punle are distinguishable by being physically unmateable; FIG.19 is a diagrammatic view showing only two game pieces of a set of interlocking game pieces having differently coded peripheral sides and usedfor assembling a checkered gamboard for checker-type or chess-type games; FIG. 20 is a diagrammatic view showing one of a set of triangular game pieces for use in a spelling game; and FIG. 21 is a diagrammatic viewshowing one of a set of pentagonal game pieces for use in a spelling game. .Referring to the drawings, and particularly to FIG. 1, there is illustrated a portion of a game or game set 10 which includes a plurality of triangularly shaped bodies or game pieces 12 and 14, each game piece having a thickness to facilitate its use in game play, and being made of cardboard, wood, ceramic, plastic, or the like, as desired (the particular material not forming a part of the present invention). , For purposes of clarity in explaining the unique mechanics of this invention, as well as for ease in illustration, the distinguishable sides or bounding edges 15 of the game pieces are type v indicated by identifying means or indicia, such as the letters A,B,C,D,E,F,G,I-I, generally designated 16, although in actuality sensorily perceptible forms will be used as explained below. Generally, sensorily perceptible" forms are considered as being best for peripheral side indicia and are meant to be inherently alike, as colors, shapes or symbols which can be matched like to like. The learned associations of sequence, as 1, 2, 3, etc. or A,B,C, etc. or Ace, King, Queen, etc. are so much a part of the game art that they might also be included, but they are considered as being better suited for use as center portion identity elements. Any two game pieces 12 and 14 constitute the simplest formulated set of distinguishable enantiomorphous game pieces wherein the distinguishable sides 15, with their identifying means 16, can be arranged in only two different and distinguishable ways to outline two different triangular game pieces, one being themirror image, or enantiomorphous arrangement, of the other. If one of the game pieces 12 is placed in the center of a table, observers at the three sides of the game piece will see the sides 15 as identified as ABC, BCA, and CAB, it being noted that the sides A, B, C have been arranged clockwise on the game piece, which convention will be observed throughout. If the same letters in the reverse order, C,B,A, are disposed clockwise as on game piece 14, the same observers will see the sides as CBA, BAC, and ACB. In the first instance, ABC, BCA, and CAB are permutations of ABC, although they are indistinguishable from each other for purposes of this invention, since they are all found on the same game piece. In the second instance CBA, BAC, and ACB are additional permutations of ABC that also are indistinguishable from each other since they, too, are found on a single game piece, but they are distinguishable from those of the first instance. Thus, there are two distinguishable arrangements for the game piece sides, and they are ABC and CBA, each being the mirror image, or enantiomorphous arrangement, of the other, which can be easily checked with amirror. When a number of these sets, or pairs, of game pieces 12 and 14 are used together, as in the group 10 of FIG. 1, the property of limited mating according to position in two-dimensional assembly, herein termed unpredictably limited compatibility, is demonstrated. It is noted that a game piece 12 cannot be mated in the open position designated at 20 since its sides B and C are the reverse of the order required at 20, nor can a game piece 14 be mated in the open position 18 since its sides A and C are the reverse of the order required at 18, while neither 12 nor 14 has the two B sides required to mate in the open position at 22. Even though each of the game pieces 12 and 14 in FIG. 1 is constituted of a different arrangement of sides of the same kind, and no interference to matching due to the symbol differences characteristic of dominoes and the playing pieces of the prior art is apparent. Yet it is obvious that mating can be positionally limited in the two-dimensional assembly of the plurality of game pieces that uses the simplest set of mathematical relationships of this invention. Incompatibility between playing pieces in the games of the prior art is based on the incompatibilities that are definitive of mathematical combinations, since it was not recognized that two-dimensional assembly creates positional incompatibilities. Neither were permutations nor pluralities of sets used to make larger games because incompatibilities of position were not recognized. It is noted here particularly that a plurality of any combinational set not only makes a larger game, as expected, but it also creates a partial set of permutations with new characteristics. These new characteristics are further enhanced and carried to their ultimate development in a certain class of permutations herein used and described as distinguishable enantiomorphous arrangements of the different bounding sides of the game pieces of this invention. Further explanations will be aided by an examination of the formulas of this invention; The general formula previously shown as: ABC .n +An CB shows how the distinguishable enantiomorphous arrangements of the game piece sides of any game piece set are formed. This general formula is written specifically for each set of game pieces of 3,4,5,6,7,8, or more sides as shown below: ABC ACB ABCD s ADCB ABCDE AEDCB ABCDEF AFEDCB ABCDEFG AGFEDCB ABCDEFGH AHGFEDCB ABC ..n An ..CB The individual letters in these formulas represent the distinguishable bounding edges that form the unique polygonal game piece bodies of this invention; each pair of letter groupings defines a unique group of enantiomorphous polygons constituted of the said sides in clockwise order, starting with the same type of initial side; and the reversible operator in each formula indicates that the unique set of pairs of polygonal bodies is so fonnulated that either member of any pair is perrnutable to the opposite and distinguishable reflection of the other member of the pair when viewed from any of the like sides of either one; and, further, that the complete set of such enantiomorphous pairs for any polygon is formed by permuting successively the symbols on one side of the operator such that each mirrored arrangement of symbols representing the sides of a ploygon generated on the other side of the operator by such permuting is a distinguishable arrangement of the respective polygonal sides; and still further, that when the number n of different sides exceeds the rate of use r on each polygon of the set, the additional symbols are substituted singly, and in all combinations, for every symbol, or combination, existing in the just-described sets of enantiomorphous permutations to create the additional members of the set. According to the present invention, the formula determines the number of game pieces in any set. The game piece sets of this invention are used singly and plurally as well as segmentally, as will be shown, wherein the unique characteristics of unpredictably limited compatibility, repetitive assemblability, and the absence of memorizable peculiarities are present. Specific sets are conveniently referred to by the ratio m/r, that is, three different sides for three sided game pieces become respectively 3/3, while four different sides for three sided game pieces becomes 4/3, and four difl'erent sides for four sided game pieces becomes 4/45, etc. The game pieces 12 and 14 of FIG. 1, as previously demonstrated, are the only two members of the 3/3 set, and the conventional solution of the formula bears this out as follows: n 3X 2X 1 2 2 (n-r) l 1' (3-3) 13 1 According to the specific form of the formula of arrangement for triangular sets, these members are ABC AC8, wherein the second member may be pennuted empirically from the first member by moving the A in ABC from first position to last to form BCA, and then reversing the order of the letters to form ACB, so that each member of the pair represents an enantiomorphous arrangement of the other, as may be proved by reflection in a mirror. In this way, both members of an enantiomorphous pair are started with indentical sides of similar game piece bodies, and proceed in clockwise order around the bounding sides of said bodies. This a method of derivation and notation is constant for all sets. The eight members of a 4/3 set, not illustrated, are formed by substituting a forth identification symbol D for each symbol in the first member of the pair constituting the set ABC r- ACE, and then pennuting the three members thus formed to create their three enantiomorphous arrangements, thereby providing the additional six members of the game set of eight asfollows: ABC ACB The basic set. ABD 4-- ADB Substitute D for C and permute the mirror image member. ADC ACD Substitute D for B and permute the mirror image member. DBC DCB Substitute D for A and permute the mirror image member. The 20 members of a 5/3 set are formed by substituting a fifth symbol E for D, and then using both D and E together as 15 DE. with A,B, and C respectively, thereby providing the additional 12 members as follows: AEC ACE ABE AEB Substitute E for D and per- Referring now to FIG. 2, there is illustrated a portion of a 30 game set 26 constituted of ten enantiomorphous game piece pairs 28, 30,32, 34, 36, 38, 40, 42, 44, and a pair 12 and 14 as illustrated in FIG. 1, wherein five n different kinds of bounding sides 16 are utilized at the rate r of three per game piece. These arrangements, conveniently referred to as the 5/3 game set of size 20 include the arrangements of both the 3/3 and the 4/3 sets plus the distinguishable enantiomorphous arrangements due to the inclusion of the fifth differently identified side. The 20 game pieces of the group 26 of FIG. 2 are illustrated with'distinguishable symbols 16 so disposed around the periphery of the grouping that any plurality of such groups can be mated to one another, thus showing the characteristic of repetitive assemblability. Referring to FIG. 3, all of the characteristics of the triangular game pieces are more evident as attributes of the square form of the game pieces of this invention. Mathematically, there are 24 permutations without repetitions, but only one combination of four things taken four at a time, while there are six distinguishable arrangements of four distinguishably different sides for square game pieces. These six unique arrangements are constituted of three unique enantiomorphous pairs as shown by the formulas for the square game pieces of this invention: =6 Distinguishable Arrangements. dditional arrangements gggg 333, as ed by .1... f1... ormu a. As a matter of physically demonstrable fact, there can be one and only onereflection, or mirror-image arrangement, for any one arrangement of physical forms. This is depicted by the formula ABCD ADCB wherein A, B, C, and D represent the different game piece sides in clockwise order in the member to the left of the operator which member can be empirically rearranged to form the mirrorimage member to the right of the operator by moving the A from first to last position as BCDA, and then reversing the order by starting with the side A to produce the mirror-image arrangement of the game piece sides ADCB. This gives two enantiomorphous arrangements, starting with the same initial side, that have the succeeding sides around the game pieces noted in clockwise order. One formula shows there are six distinguishable arrangements that the other formula indicates must consist of three enantiomorphous pairs. Mathematically, any symbol can represent any assigned identity so that in the arrangement ABCD, for example, D can represent the fourth game piece side having the assigned identity C, and also in the same arrangement ABCD, C can represent the third side having the assigned identity D. Therefore, ABCD represents the game piece whose respective sides have the assigned identities ABDC; and, in like manner, also represents the game piece ADBC; it being noticed that the symbol D is moved from fourth position to third, and then to second, to empirically form three different arrangements for the game piece sides. It is demonstrably obvious that the reflected arrangements of ABCD, ABDC, and ADBC are distinguishable for purposes of this invention so that the three enantiomorphous pairs of distinguishable arrangements are produced. In like manner, and as previously shown for triangular sets, additional sides n can be substituted by those skilled in mean to form the sets indicated by the formula it is also demonstrably obvious that if the D of the original ABCD were placed in the first position, the resultant DABC is not distinguishable from ABCD for game piece use. In like manner, each of the specific forms of the general formula ABC ..n An ..CB represents within its own order of magnitude the implicit relationships of the enantiomorphous forms of every possible distinguishable arrangement of any peripheral sequence of different game piece sides in clockwise order, as for example,0 o -i +0 wherein 0 is the initial side equivalent to A, to B, o to C, and +to D. Each distinguishable arrangement on one side of the operator is empirically convertible to the enantiomorphous form of the distinguishable arrangement on the other side of the operator, and since there is only one mirror image for any one form, the formula represents all the possible distinguishable permutations shown by the quantitative formula. All the possible forms are permuted according to mathematical convention to produce the calculated quantity. However, as required for explanatory purposes, the basic sets will be tabulated, except that the very large sets will be shown in easily extensible form. As illustrated in FIG. 3, the three enantiomorphous pairs of polygonal bodies or game pieces designated 48 and 50, 52 and 54, 56 and 58 constitute an assemblage 46, with the peripheral sides of the individual game pieces arranged to make the assembled set repetitively assemblable with similar game sets to provide games of almost any size. It will be noticed that each game piece is compatible with every other game'piece, so that no memorizable peculiarities attach to any game piece. Therefore, incompatibilities between game pieces depend on assembled relationships that are predictable only when the order and position of assembly of each game piece are specified. The order of random choice of the unique game pieces of this invention is one basis of the characteristic called unpredictably limited compatibility. Referring again to FIG. 3, and to the game piece designated 54, it will be seen that no other game piece has the indicia 16 for the sides C, A, B in that clockwise order, so that no other game piece will play in that position. If the game piece 54 were played to mate the sides A and B in the position of game piece 48, or to mate the sides C and A in the position of game piece 56, neither of the game pieces 48 or 56 would mate in the position of game piece 54. However, these conjectures presuppose an order of assembly that is only one of many possibilities. In order words, positional compatibility in twodimensional assembly also is determined by the arrangements of the distinguishable sides of each of the game pieces, by the random order of assembly, and by the reasoned choice of the assembler. Thus, skill is incorporated in playing the game pieces to form a two-dimensional game, since every game piece must be mated finally onmore than one side according to its order of play, the arrangement of its distinguishable sides, and the reasoned choice of the competing players; the said game pieces being repetitively assemblable, without memorizable peculiarities, unpredictably limited in compatibility, and statistically unbiased for game purposes. Positional compatibility is further affected by permutation sets larger than game requirements to increase the number of assembly patterns. This further conceals the order and exact content of the assemblage of game pieces until the last game piece is in place, and increases the degree of statistical impartiality to fantastic limits. Examples are the use of the 6/4 game set constituted of distinguishable game pieces from which the 64 squares of a checkerboard are randomly selected and assembled; or, the use of four of the 5/4 game sets constituted of 30 game pieces each, from which are selected the squares to assemble a ten-by-ten checkered board, it being understood that either checkered pattern would be somewhat irregular. In contrast, the playing pieces of the prior art, with the relationships of their one-set, or partial set, indicia combinations, cannot gain the statistical impartiality just described without using inordinately large sets, and without using still larger sets when indicia repetitions are included in the combinations. The playing pieces of the prior art copy the linearly compatible mathematical combinations characteristic of dominoes that are not repetitively assemblable two-dimensionally, and that do possess memorizable peculiarities based on inherent combinational differences that have obvious and predictable incompatibilities. Further, it must be remembered that duplicate sets of combinations make a game not only obviously larger but unexpectedly different because duplicate sets of combinations, constituting the game pieces of the same game, become permutations by definition, and the unexpected characteristic of unpredictably limited positional compatibility exposed thereby is a discovery and development of this invention. The physical similarities and arrangements of shape, color, or symbol as integral parts of the game pieces of this invention are obvious in varying degrees to untutored natural ability, and they exist independently of game intelligibilities dependent on the learned symbolism of formal education that are applied to the game pieces to make difierent games. The following examples incorporate game intelligibilities to further demonstrate some of the applications of the game pieces of this invention. Referring to FIG. 20, a triangular game piece 270 with colored peripheral indicia 272 and having a letter 274 of the alphabet for use in a spelling game is illustrated. 'Referring again to FIG. 3 and to game indicia or intelligibilities 60 and 62 which occupy the center portions of the game pieces 48, 50, 52, 54, 56, and 58, if, for instance, a spelling game is the intended use, the L on game piece 54 can be used as the initial letter of the word lull by passing in order across the game pieces 52, 48, and 50; or lulu across 52,- 56, and 58; or any word containing LL such as llama or ball by crossing game piece 50. The illustration of FIG. 3 is understood to include only a few of the game pieces of the game. A player of such a spelling game is confronted with assembling intelligible letter relationships that also must conform to game piece mating compatibility. This dual, or coincident, compatibility requirement is missing from the simple letter cards for spelling, and is present to some fixed degree in spelling games that utilize a game board, but the game pieces of this invention combine the simplicity of the letter card with the complications of the game board in a variable manner that is new because there are no builtin patterns and no memorizable peculiarities. If a chess or checker-type game is the intended use, referring again to FIG. 3 and the game piece intelligibilities 60 and 62 wherein L represents one color and U represents the second color of the squares of a checkered board for purposes of this example, two squares of the same color are illustrated in adjacent positions. It is understood that the illustration includes only a few of the game pieces of the game and that an irregularly checkered board is to be assembled. The assembly of the many differently arranged checkered boards constructed in this manner is subject to the will of the players, with the result that completely new strategies replace the traditional strategies of chess and checkers, thus creating new games. should not be present on playing pieces having no point of similarity for mating to one another. For example, the intelligibilites that might be added onto domino double-one should not be related to the intelligibilites added onto domino douif a travel-type game is the intended use, it is obvious that ble-two. The 5/3 triangular set of combinations with repetithe intelligibilities on the faces of the game pieces of the previtions of the prior art is limited mathematically to 35 playing ously mentioned travel-type games of this invention require pieces; the 6/3 set is limited to 56 playing pieces; and the 9/3 knowledge and skill to relate these game piece face intelligiset to 165. When the number of different game piece sides bilities for proper assembly, and that each conventional game equals twice the rate of use per game piece n 2r), the total played on the surface of each differently assembled surface 10 incompatibilities represented by ABC to DEF are created. in can be somewhat different. in other words, any travel-type addition, there are the easily memorized relationships of the game constituted of the game pieces of this invention really is intelligibilites to the double and triple repetitions represented a large plurality of games, each possessing elements of skill. by AAA, AAB, and AAC. it is, therefore, mathematically im- Of course, the travel-type game that relates geographical possible for the combinations of the prior art to gain a sufiit'acts in game piece assembly constitutes the model for all sorts cient quantity of the kind" of differences required to imparof educational games wherein the facts, or suppositions, of histially differentiate the game pieces required for many games. tory, science, economics, or any other past or present en- Statistical bias is present in the 5/3 set of combinations of dcavor can be related in assembly and scored for a variety of the prior art until all 26 letters of the alphabet are present on educational games, as will be evident to those skilled in the each of the 35 different playing pieces, for a game size of (26 art. The many different multisided game pieces,aswell asmix- X 910 playing pieces. However, the duplicated comtures of different multisided game pieces, provide the possibinations are permutations whose applications are the bilities for associating in assembly the various degrees of pludiscovery and development of this invention. ral relationships that generally exist between the significant The tabulations of formulated set sizes of various (n/r) events of human endeavor. 2 ratios for prior art playing pieces, for sets of interest, and for It is obvious that intelligibilites related to one another 5 someof the game pieces of this invention are as follows: gara- Combina- Permuenari tlo tions tations morphous without Permuwithout arrange- -n/r repetitions Combinations with repetitions tations repetitions merits (Prior Art) 21 28dominoes dominoes-.. 78 91 dominoes. 1 1 27 4 20 e4 10 35 Richards 125 20 66 Richards 216 84 165 Palmer and Meili e 729 1 35 256 5 70.. 3,125 15 126. 35 210... 1 126..- 3,125 6 252.-- 1 462..." 46, 626 1 1,716 1 6,435 Formulas w n! (n+r-1)! 'n' n! n! y n-nm (n1)lrl (n-r)! (n--r)!r e (1.8. Letters Patent No. 331,652, Frank H. Richards. b U.S. Letters Patent No. 2,072,605, Paul Louis Palmer. British Letters Patent N 0. 6,016 filed April 21, 1890, August Meili. NOTE: Patterns-Combinations with repetitions: EABC' n-i-ABC n Patterns-Distinguishabie enantlomorphous arrangements: gABC 1t oAn CB Distinguishablc permutations with distinguishable Formulas repetitions Quantity Pattern N'lE.-SYMBOL DEFINITIONS: 1|, is the number of peripheral-side identity-elements of any selected setgroup. A, B, C, etc designate the periheral-side identity-elements (color, shape, etc.). 1, is the characteristic number of peripheral sides of the selected polygonal bodies. It is used when a formula covers more than one characteristic shape. n r, symbolizes that "n is never less than r (to avoid negative numbers). ABC n, means to combine (as shown later) the selected number of diflerent peripheral-side identity elements into all the peripherally distinguishable combinations (mathematical) possible "r" at a time without using the said peripheralside identity-elements repetitiously. (When the formula does not include "r", it applies to a specific type of polygonal shape and the number of capital letters is the same as the number of peripheral sides, thus showing that each set is built upon the preceding set of its type.). . n means the same as the above but, for clarity is used when the formula contains the collective r (even though the BC -triangularABC n" is evident in context). ABC n means the same as the above except that only the distinguishable peripheral indicia members using the said per- R ipheral indieia-elements repetitiously are included (the underline is not doubled). This operator (mathematical operation indicator) indicates the sequence reversal of the so-ioined peripheral identityelcments combinations (mathematical) to form distinguishable arrangements (permutations). Note that this operation reverses the arrangements shown above. it indicates the class of all peripherally distinguishable peripheral combinations (mathematical) including those using the said R peripheral-side identity-elements repetitiously. (Note that"ABO a" andABC 1 r" are sometimes used redun- R dantly with if (or iormula standardization and ior clarity, but the result of such redundant use is equivalent to adding zer0.). C, indicates the class of all peripherally distinguishable peripheral combinations (mathematical). (The bar under the C" is to distinguish mathematical eombinations" from the sides 0" even though the difference is evident in context.). Pa, indicates the class of all peripherally distinguishable peripheral arrangements (permutations) without the identity-elements being used repetitiously. 3 4, 5, 6, 8, indicate the characteristicnumber oi the selected peripheral sides. 2 Indicates the summation oi all oi the types n, C",P",r" or3",4",5",6" and8") shown along the bottom leg R oi the said sign in accordance with the patterns following the said sign. Explanations for the following described typical sets of game pieces of this invention can be brief, in view of the prior explanations. Y The preceding tabulation shows that the 5/5 pentagonal set is constituted of 24 distinguishable arrangements of five different kinds of game piece sides, and the 6/5 set has 144 distinguishable arrangements. Regular pentagonal shapes are not physically assemblable to completely cover a plane surface, but the comparative sides will match across the diamond-shaped voids created by the geometry of the assembly so that each pentagonal set is relatively repetitively assemblablc. Because of the voids, some game paths across the surface of bodies of assembled pentagonal game pieces are interestingly discontinuous. f m Present in all the sets of this invention, but particularly obvious in the larger sets, are subsets that possess all the characteristics previously described. The following table shows the members of the 5/5 set with the 12 pairs of arrangements numbered one through 12, wherein both members of each enantiomorphous pair have the same number with one of the numbers being bracketed in parentheses. The basic 24 members are shown in four columns of three pairs of lines constituting six groups of four. Some of the many repetitively assemblable subsets are constituted of the members shown in any column, or any row, or any group of four The eight members appearing in the last two lines are utilized in FIG. 4, in which is illustrated a repetitively assemblable group 64 constitutedof pentagonal game pieces designated 66, 68, 70, 72, 74, 76, 78, and 80 shown without game intelligibilities for clarity. Applications of the pentagonal game pieces to spelling, travel-type, educational, and other games are obvious to those skilled in the art from previous discussions. ' Referring to FIG. 21, a pentagonal game piece 280 with colored peripheral indicia 282 and having a letter 284 of the alphabet for use in a spelling game is illustrated. The 6/6 hexagonal game set of this invention has 120 members constituted of 60 enantiomorphous pairs. The set and the subsets possess all the usual characteristics previously described for all the sets. The following tabulation of the members of the set is arranged to show some of the ways the members can be grouped into subsets. The members in any line from columns 1 and 2, 2 and 3, 3 and 4, 4 and 1, or the entire line are repetitively assemblable subsets. lt is possible to accidentally duplicate small portions of the sets of arrangements utilized in the game pieces of this invention, but the number of games to which such portions might be applicable is severely limited, and none are known to exist in prior art. Had any of the sets of distinguishable enantiomorphous arrangements been developed accidentally and used for games, the advantages would have lead inevitably to mathematical investigation and resulted in their derivation and inclusion in the allied literature of mathematics and games. Since this has not happened, it is desirable to show how all the sets of arrangements for the sides of the game pieces of this invention are developed. It has been shown that the permutations of three things, taken three at a time, or 3/3, is denoted by the arrangements: ABC, BCA, CAB, and BAG, AOB, CBA. 1 ABC A013 (1), the operator indicating that each arrangement is the enantiomorphous form of the other. The derivation of the customary set of permutations from the enantiomorphous set is obvious, just as it is in all the sets, and there is no record of the anticipatory use of permutations in games. 1 ABCDEF 29 ABFDEC 59 AEFDBC 2 ABCDFE 32 ABEDFC (i AFEDBC G ABCEDF 34 ABFEDC 54 ADFEBC 7 ABCEFD 27 ABDEFC 55 AFDEBC 3 ABCFDE 33 ABEFDC 53 ADEFBC 8 ABCFED 28 ABDFEC 58 AEDFBC 21 ABDCEF 9 ABFCED (42) AEFCBD 22 ABDCFE 12 ABECFD (41) AFECBD 26 ABDECF 14 ABFECD (47) ACFEBD 23 ABDFCE 13 ABEFCD (48) ACEFBD 11 ABECDF 4 ABFCDE (17) ADFCBE 31 ABEDCF 24 ABFDCE (37) ACFDBE (40) ACDBEF (52) ACFBED AEFBCD (39) ACDBFE (49) ACEBFD 2O AFEBCD (34) ACDEFB (54) ACBEFD AFBECD (33) ACDFEB (53) ACBFED 18 AEBFCD (50) ACEBDF (57) ACFBDE 44 ADFBOE (2U) ACEDFB (59) ACBDFE 25 AFBDCE (28) ACEFDB (58) ACBFDE 43 ADBFCE (32) ACFDEB (60) ACBDEF 36 AEBDCF (27) ACFEDB (55) ACBEDF 46 ADBECF (14) ADCEFB 47 ADBEFC AFBEDC (13) ADCFEB 48 ADBFEC 38 AEBFDC 52 ADEBFC (19) ADCBFE (56) AFCBDE ADFBEC (20) ADCBEF AECBDF (12) ADFCEB 41 ADBCEF 16 AEBCDF 57 AEDBFC (44) AECBFD (51) AFCBED (4) AEDCFB 17 AEBCFD 10 AFBCED 5 AFBCDE (21) AFECDB (9) ADECFB 30 AFBDEC (31) AFCDEB (24) AECDFB (30) ACEDBF 56 AEDBCF AFDBCE (23) AECFDB AFEBDC AEFBDC (11) AFDCEB ADBCFE AEBDFC The 12 members of the lines beginning with the numbers 1, 8, and 21 are utilized in FIG. 5, in which is illustrated a repetitively assemblable group of hexagonal game pieces designated 84, 86, 88, 90, 92, 94, 96, 98,100, 102, 104, and 106, shown without game intelligibilities for clarity. Game applications will be obvious to those skilled in the art. . However, one such application of interest is shown in FIG. 6 wherein is illustrated a portion of a game 108 constituted of hexagonal game pieces designated 110, 112, 114, 116, and 118 that incorporate an hexagonal subset, and that also show game intelligibilities 120 consisting of the usual identification used in a deck of playing cards. Many rules may be formulated forplaying games related to Rummy and Poker with the justdescribed game pieces, but for the purpose of brevity, the playing rules are omitted, particularly since the availability of such an arrangement of hexagonal game pieces will immediately suggest suitable rules to those skilled in the art. The 4/3 set of four different sides A, B, C, and D, used three at a time, wherein n is greater than r was previously shown as: 3/3 1. 4/3 1 AB CB (1) 2 A A (2) 3 C C A (3) 4 BC CB (4) the sets in which n is greater than r. The significances are marked as before: 3/3 4/4 1 ABC A CB. 2 AB C AC B 3 A BC ACB -The /4 set uses the six arrangements of the 4/4 basic set and the arrangements formed by substituting a fifth side E for each of the sides A, B, C, and D, marked as before: The 6/4 set of 90 members requires the double substitution previously shown for the 5/3 set. The 30 members of the 5/4 set, that utilize the sides A, B, C, D, and E, are included in the 6/4 set, plus the 24 members that can be fonned by substituting a sixth side F for the side E in the 5/4 set, plus the 36 members that can be formed by using EF together and then separated (-FF) with all the different pairs that can be chosen from the sides A, B, C, and D to enclose a four-sided game piece. One member of each pair is constituted as shown, the other member being perversely arranged as explained earlier or tabulated in the manner indicated: 28 AEFB BFEA (28) 29 C CFEA 30 D DFEA 31 EBF FBEA 32 C FCEA 33 D FDEA 34 BEFA "AFEB 35 C CFEB 36 D DFEB (36) 37 ECF FCEB 38 D FDEB 39 CEFA AFEC 40 r B BFEC 41 D DFEC 42 EDF FDEC 43 DEFA AFED 44 B BFED 45 C CFED (45) I Returning now to h=r sets, the 5/5 set is developedby positionally adding a fifth side E to the preceding set of the same The 6/5 set is constituted of the 24 members of the 5/5 set plus the l members formed by substituting F for each of the sides A, B, C, D, and E in every member of the 5/5 set, as previously shown for sets wherein n is one greater than r. 5 The 6/6 set, based on the 515 set, follows the usual pattern: Both the septagonal and octagonal game pieces possess geometrical voids in assembly that produce game path discontinuities, as do all the other sets of higher order develpped by a continuation of the arrangements. The septagonal form, in- cluded here for continuity to the following octagonaharrangements, is based on adding a seventh side G to the 6/6 set: The octagonal arrangements are based on adding an eighth side H to the 7/7 set, partially shown above: v 7 A BCDEFG AGFEDCB 7/7 2. 8 ABCDEG A FGEDCB 9 ABCDE F The 7/6, 8/7, and 9/8 sets have 840, 5,760, and 45,360 members, respectively, and are developed in the manner of the sets wherein n is one greater than r. They are much too large for normal use, but some of their subsets are useful. Some of the 7/6 subsets are applicable to the playing card type of game illustrated previously. The 9/8 and the 8/8 subsets are applicable to games played with the octagonal game pieces of this invention. Referring to FIG. 17, a portion of a playing card type of game 220 is illustrated wherein the octagonal game pieces 222, 224, 226, 228, 230, 232, 234, 236 and 238 have playing card type: center indicia 240 and peripheral indicia 242. Preferred paths 244 are shown across some of the game pieces to illustrate still another variation. Many games somewhat resembling Rummy, Poker, or Travel games will be ap- 75 parent to those skilled in the art. it is obvious, at this point, that the rotation of the members of the sets denoted by (nr)!r 11.! produces all the sets denoted by and that all the sets denoted by (nr) 11'! are delineated by the first inclusion of elements that Prior art used only sets of combinations with repetitions of the form wherein r did not exceed 3, and for square playing pieces wherein it did not exceed 3. In no case has prior art used pluralities of sets, nor have any of the sets possessed the characteristics of repetitive assemblability and unpredictably limited compatibility along with the absence of memorizable peculiarities, so that no statistical bias pertains to any game piece, all as defined herein. The prior art failed to produce more than triangulalfl dominoes, square puzzles, and a truncated-triangle spelling game based on combinations with repetitions, so that no combination of the prior art will produce the assemblable gameboard or is suitable for application to the well-known game forms, as are the sets of this invention. Those skilled in the game art can develop many games with the game pieces that utilize the formulated relationships just described. However, these contrived mathematical relationidentical to one another, as were the segments with the projection 124 and the indentation 126 discussed above, while the diagonally opposed segments of the androgynously mated sides of the two game pieces are identical. The remaining sides of the game pieces 122 of FIG. 7 show the indentation 132, located on the centerline nonnal to that side of the game piece. The projection [34, superposed on the centerline of the side and outlined by the cut that separates the two game pieces 122, really is an undivided pair of identical projections disposed 180 to each other in the same plane. In this case it is obvious that the indentations are the inverse identity of the projections, since the same cut delineates both of them, yet does not divide the projections. As before, the configuration of the segment disposed on one side of the centerline of one side of one game piece is identical to the configuration of the segment of the side of the other game piece disposed on the other side of the center line, while the two game pieces are in the mated position and the two said centerline are coincident. The unique androgynous mating of the polygonal shapes of this invention is identifiable and exists if the outlines of the configuration and the extended centerline of each member of a mating pair of peripheral sides of two polygons are respectively congruent when placed side by side and when superposed after one polygon is rotated 180 in a parallel plane. Earlier it was shown that the top and bottom surfaces of such androgynous sides always are minor images of one another. All this is unique with androgynous mating, regardless of the variations of size, shape, or relative positioning that may be devised by those skilled in the art. If the projection on any side is symbolized by (69 M) and its inverse shape in the form of an indentation by 9 F), the top and bottom faces of an entire peripheral side can be denoted by 2% with the center point shown as a centered dot. By utilizing the operator denoting the enantiomorphic form, the androgynous relationship of the top faces to top faces and bottom faces to bottom faces of two mating sides may be represented by: IGMSF @MGFI QBMGF GBMGF wherein is indicated the male-famle, or plus-minus, fonn of such androgynous relationship. When the indicated game pieces are rotated to mate side-by-side or face-to-face, it will be seen that each 69 M) mates to a (9 F). if one member is reversed as ships can be 8 P y identifies that are always which is identicaltothe result of turning a game piece identifiable by the characteristic positioning of the parts regardless of the particular form. FIG. 7 shows obviously congruous game pieces 122 juxtaposed l80 to one another so that a bounding side of one, that has both a projection 124 and an inversely identical indentation 126, is adjacent to the one bounding side of another that is constituted with an identical indentation 126 and an identical projection 124, the projection 124 on the said side of each game piece being disposed similarly to the indentation 126 of the other game piece with respect to the centerline normal to the said sides, so that the outlines of the two sides are congruent in the mating position. It is demonstrably obvious that such identical sides of any two identical game pieces are physically matable, so that they may be analogously termed androgynous, or inseparably bipolar. Again referring to FIG. 7, the projections 128 and indentations 130 make two other sides of each game piece respectively matable androgynously. The configurations of the two segments, disposed each side of the centerline nonnal to one androgynously matable side of each game piece, are inversely upside-down, the physical mating of G9 M to 6 F between game pieces becomes impossible. Thus is demonstrated in symbolic form the precise mathematical relationship previously described. The extension of the physical form of androgynous mating to include interlocking is illustrated in FIG. 7 by the use of ordinary dovetail projections 124 and indentations 126 that easily engaged yet resist separation in the plane of use to own destruction. An equally common solution for interlocking -partif'is 'tlie' fi'ict'ional'fit between'the sideso'fprojections an indentations, of which those at 128 and l30are representative. Also, magnets are peculiarly adapted to hold together the sides of the separate polygonal pieces of this invention, since every magnet is constituted of two different but inseparable poles analogous to the androgynous form. The game pieces 122 of FIG. 7 are identical, and each pair of identically configured sides are matable, remembering, of course, that the projection piece 134 is a joined pair of projections constituted of the parts of two game pieces 122 as outlined by their adjoining indentations 132. The projection piece game piece 122 represent only a few of the possible variations of shape, size, and spacing identifiable as the androgynous mating of this invention. It is also obvious that the sequential arrangement of the different sides that constitute the periphery of each game piece is subject to change. If the differences of shape, size; and relative position with respect to the centerline between the geometrically inverse projections and indentations are symbolized by X, then X M 9X F EBX M 6X F denotes every peripheral side difference indicated by any ar rangement of (ABC ..n), and the formula for every physi- V cally mating pair of peripheral sides is: The application of this symbology to the several forms illus trated in FIG. 7 is obvious, while the quantity and composition ,of such sets are as previously described. The prior art has not utilized the androgynous configuration for mating playing piece sides, and, while permutationshave been available at least since Bemoullis Ars Conjectandi" was published circa I715, their application for the sequential arrangement of game piece sides is'unique with the game ,pieces of this invention. The permuted arrangements of the sides of game pieces allows for mating many different assemblies, and the androgynous interlock ties any possible assembly together to create assemblable gameboards. The jig-saw puzzle ,is the closest approach in the game art to the interlocking game pieces of this invention, and the differences are definitive. Jig-saw puzzle pieces are the inverse of one another in the sense of male andfemale, and any differentiating label given to the physical identity of the female configuration obviously cannot be the same one given its matable male. configuration, but the mating sides of the polygonal game pieces of this invention are identical and, hence, are identically indicated so that shape A mates to its identical shape A, not to a complementary shape B. Jig-saw puzzles by nature and definition are puzzles wherein there exists only one solution or physical assemblage which is analogous to the shape relation ships of the puzzle pieces, since the playing pieces are not irite -.hangeable with one another. Indeed, the shapes of the puzzle pieces and-the knowledge of the picture that was cut into the puzzle piecesprovide the clues and the limitations for the one possible and invariable assemblage. r The sets of indicia combinations of the playing pieces of the prior artare as inflexible for different aggregations as either the picture or the shapes of the jig-saw puzzle. In fact, the differences-characteristic of jig-saw puzzle pieces are somewhat analogous to the differences inherent in the sets of combinations used as indicia for the playing pieces of the prior art, in that only one possible assemblage exists, just as the similarities of the androgynous configurations of this invention are somewhat analogous to the similarities of permutations whereby many different arrangements are possible in assembly. Jig-saw puzzles, as well as the straight sided polygonal play- pieces of the prior art, have twofaces available for use but wrong face in the upward position obviously is mechanically incompatible with the other pieces having the right face in the upward position, since the top and bottom faces of the androgynous shape are unmatable mirror images of one another. FIG. 18 shows the top and bottom faces of one game piece 250 having difierent pictures 252 and 254 on its faces and illustrative of the jigsaw puzzle pieces of this invention as just discussed. Referring now to FIG. 19, there are shown two different game pieces 260 and 262, one having a black central indentity 266 and the other having a white central identity 268, while both have peripheral identities 264 comprised of color for quick visual identification, and shape for interlocking into a checkered board for checker-type and chess-type games. It will be obvious to those skilled in the art that these central identities can be any of the identities commonly used in board games to produce not only many different board games, but also many variations of such games, depending on the competitive assembly of the assembleable gameboard by the players of the game. As previously explained, the term distinguishable mirrorimage arrangement" indicates the reversed order of the distinguishably different sides around the periphery of the polygonal pieces. The related term distinguishable mirror image" is a mathematically precise term analogous to the mirrored differences, for example, between right and left hands, and herein refers particularly to the distinguishable differences between the top and bottom face outlines of the androgynous sides of the game pieces of this invention. The term androgynous, as used herein, refers to the combination of male and female attributes in one identity somewhat like the two inseparable poles of a magnet and, therefore, by definition, each androgynous game piece side is constituted of at least two essentially different but related parts, combined to create a single inseparable identity. It is demonstrably obvious that the arrangement of the parts of androgeneity is reversed in the mirror image so that no possible rotation in a single plane will permit the inseparable parts of the androgeneity to mate with their own distinguishable mirror image. In contradistinction, a straight line side of a In other words, the androgynous identity is a contrivance constituted'of two (or more) different parts, as previously defined, which combine into the unique identitythat will mate to its own identity, but not to its own distinguishable mirror image. It is particularly important to note at this point that mating the true androgynous identity requires no acquired knowledge of language, number, or mathematics by the opera- I tor; I Referring to FIGS. 8 and 9, there is shown an assemblage constituted of six square game pieces designated 152, 154, 156, 158, 160, and 162 arranged in FIG. 9 as a flat pattern that schematically shows the relationships of the six faces of a cube. Each cube facehas the letters M, designated 164, and the letters F, designated 166, located alternately around its four peripheral sides. For purposes of explanation, M and F are considered to denote the characteristics of male an female that may penetrateentirely through the thickness of each of the square polygonal pieces, so that the top and bottom faces of each of the said pieces are distinguishable mirror images of one another. This'mirror-image form of assemblage 150 is not separately illustrated, since earlier explanations render the form obvious. The identical M and F pairs on each peripheral side of eac cube face of assemblage 150 schematically are the androgynous form previously defined, wherein contiguous peripheral sides androgynously match the two different elements, M to F; superposed peripheral sides are identical, and the top and bottom faces of the flat pattern are androgynously unmatable minor images of one another. The square polygonal cube faces 152, 154, I56, 158, 160, and 162 of the flat pattern of FIG. 9 are foldable along the bend lines, in a downward manner with respect to the plane of the pattern, so that the cube of FIG. 8 is fonned with the labeled faces outwardly exposed, and so that each androgynous duality (M and F) is paired, M's to P8, with another androgynous identity at each vertex of contiguous sides, somewhat analogous to the pairing of male and female elements, or to the pairing of opposite magnetic poles. Replicas of such cubes are androgynously matable and repetitively as semblable without limit. The said faces of the said pattern of FIG. 9 are also foldable along the bend lines in an upward manner so that the inside surface of the just-described cube becomes the outside surface of a different cube that is the mirror image of the first cube. It is obvious that replicas of the mirror image cube are androgynously matable and repetitively assemblable also. The two different cubes thus formed are not androgynously matable to one another since the said cubes are mirror images of one another, and, as shown earlier, mirror images are androgynously unmatable. However, the requisites for puzzles and games exist between the extremes of unmatability and unlimited matability just described. This conflict is resolved when the square distinguishable arrangements, previously described, are used to lend unique identity to the faces of otherwise identical cubes, since pluralities of such cubes acquire the unpredictably limited compatibility analogous to that of the game pieces of the assemblable gameboard also previously described. The 4/4 set of distinguishable enantiomorphous arrangements, applied to the cube faces, is shown in FIG. 9 wherein the letters A, B, C, and D, designated 16, are placed to indicate that each of the six top faces is a unique androgynous -=identity; and that the four peripheral sides of each of the said faces are different androgynous identities. Since three of the labeled cube faces of the cube thus created are mirror image arrangements of the other three faces in accordance with the fonn of the distinguishable enantiomorphous arrangements, half of such a cube is the mirror image arrangement of the other half. It is, of course, impossible to create this unique cube with the combinations utilized exclusively in the prior art. Such polyhedra match one another face-to-face in the same manner that every right or left hand matches exactly its own image in a mirror, with the sense of the mirror image being that of the arrangement of the opposite hand. It will be noticed that this arrangement is that of the order of the polygonal sides, while the arrangement of the individual androgynous sides is not changed, so that androgynous mating is limited by analogous to the external forms of crystals that are themselves uniquely assemblable mathematically and mechanically into assemblages. While cubes are used illustratively in describing certain mechanical features of this invention, it will be obvious to those skilled in the art that the same principles will apply to all the regular and irregular polyhedra both singly and in aggregations, just as the mathematical arrangements previously described are similarly applicable. 10 The previously described androgynous mating v for gameboard game pieces satisfies the mathematical and physical conditions of rotation in one plan, while the three-dimensional androgynous interlock satisfies the plural conditions of multiple rotations and displacements in three-dimensional space. FIGS. 10, 11, and 12 show a polygonal body designated 170, having a thickness designated 180, characteristically and appropriately beveled as shown at 182 in FIGS. 11 and 12, for mechanical juxtapositioning as one face of a cubic shell, other face pieces being designated 174, 176, and 178, the top and bottom face pieces being omitted. A three-dimensional segment of the beveled portion of a peripheral side is illustratively excised to create the indentation designated 184, and reattached to create the projection designated 186, in a relative position on the opposite side of, and equidistant from, the perpendicular bisector of the said peripheral side. This relative repositioning is the result of a shift and 180 rotation made as though the said segment were hinged at the line of juncture of the said bisector with the surface to which the projection is attached. This is consistent with the two-dimensional androgynous mating previously described, since mating sides are identical, and the configurations located on opposite sides of the said bisector are inverses of one another. Three-dimensional projections and their reciprocally inverse indentations are symbolized for descriptive purposes as (9 @l) and (9 respectively, while the thickness of any threedimensional peripheral bol. If the three-dimensional differences of shape, size, and relative position with respect to the said bisector located between the said reciprocally inverse three-dimensional projections and indentations are symbolized by X, then denotes the identity and peripheral position of the elements of the distinguishable enantiomorphous arrangements. Three dimensional aggregations of pluralities of such cubes have unpredictably limited compatibility analogous to that of two-dimensional game pieces previously described. The application of the 5/4 set creates five unique cubes with pluralities that are more diflicultly assemblable in aggregations, while the larger sets create still more unique cubes with still greater difficulties of assembly. The triangular and pentagonal sets of distinguishable enantiomorphous arrangements, previously described, are uniquely applicable to the shells of tetrahedrons, octahedrons, dodecahedrons, and icosahedrons in the manner just described for hexahedrons. The said sets together with the hexagonal, septagonal, octagonal, and square sets are also applicable to the shells of mixed polyhedra, deformed polyhedra, and mixed and deformed polyhedra. The unique mathematical relationships of the sets of permutations, herein termed "distinguishable enantiomorphous arrangements, are coupled with the unique androgynous configurations of this invention to fonn three-dimensional shells every peripheral side difference indicated by any possible arrangement of the differences (ABC....n), while the formula for every androgynously mating pair of peripheral sides of the polygonal face pieces of individual polygonal shells is



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