Title:
Pseudo four dimensional dice and game
United States Patent 3904208


Abstract:
Dice games are disclosed in which each die has a three dimensional configuration conceptually representative of a four dimensional system consisting of the usual three spatial dimensions together with a fourth dimension such as time. The time dimension corresponds to movement of a three dimensional object, such as a cube, along a time axis. One or more of these dice may be used in the game, each in the shape of a dodecahedron having 12 rhombic shaped faces and representing the four dimensional system in three dimensions much as a three dimensional cube can be pictorially represented in a two dimensional drawing. Each of the 12 faces of each die are provided with number and color indicia means for distinguishing the various faces and providing the probability determining events characteristic of dice. The face numbering and color coding system is derived from the four dimensional concept used in creating each die. The die may be used individually or together, or as disclosed herein in combination with a game board having betting zones provided thereon by which a player can select a bet based on the numbers or colors or both represented on the dice faces.



Inventors:
GROSSMAN JACK J
Application Number:
05/359773
Publication Date:
09/09/1975
Filing Date:
05/14/1973
Assignee:
GROSSMAN; JACK J.
Primary Class:
Other Classes:
273/274, D21/373
International Classes:
A63F9/04; A63F3/02; (IPC1-7): A63F3/00; A63F9/04
Field of Search:
273/146,130
View Patent Images:
US Patent References:
3208754Dice game with a tetrahedron die1965-09-28Sieve
1517113Baseball die1924-11-25Greene
0645112N/A1900-03-13



Other References:

Scientific American; Mathematical Games, Nov. 1966, Vol. 215, No. 5, pp. 138-143, by Martin Gardner..
Primary Examiner:
Pinkham, Richard C.
Assistant Examiner:
Taylor, Joseph R.
Attorney, Agent or Firm:
Miketta, Glenny, Poms & Smith
Claims:
I claim

1. A game comprising:

Description:
BACKGROUND

In general, the present invention relates to games and educational toys and in particular to games and toys based on the use of dice.

The history of dice games and related games of chance extends back to ancient times. The Latin word for dice "tesserae" is derived from the Greek "tesseres" and Ionic "tessares", meaning four, corresponding to the four edges of the square comprising each side of the standard die. The classic form of the die is a cube with six square faces, 12 edges and eight corners. Since the ancient origin of the dice game, many variations on this basic die have been derived.

For example, U.S. Pat. Nos. 1,517,113; 3,208,754; and 3,399,897 show several variations on the basic or standard game of dice. In general, a wide variety of regular and irregular polyhedra have been proposed for use as the probability determining element of a dice game. Dice or other three dimensional objects are variously used to define word games, baseball games, horseracing, card games including poker, etc. In these prior games, the probability determining devices are representative of only two and three dimensional shapes or objects.

SUMMARY

The object of this invention is to provide a dice game in which the die or dice represent a multi-dimensional concept or system having more than three dimensions. Starting with a multi-dimensional system, such as a four dimensional system, dice may be created by the use of formulae disclosed herein to reduce the four dimensions to three dimensional solid representations. This reduction to three dimensional form is much like the pictorial representation of a three dimensional cube in a two dimensional picture.

In one particular and preferred embodiment as disclosed herein, the four dimensional system in accordance with the present invention is reduced to a die having the three dimensional shape of a dodecahedron defining 12 individual die faces. These 12 faces are systematically numbered in accordance with a formula derived by reducing the four dimensional system to a representative three dimensional solid object, such as the dodecahedron.

In order to derive the formula for this numbering system, it is shown herein that the dodecahedron created by the four dimensional system is in fact composed of eight individual, but interrelated cubes, each having its own six faces. These individual faces of each of the eight cubes comprising the dodecahedron may be identified such as with the numbers 1 through 6, in a manner similar to the numbering of the six sides of a standard cubic die. However, as more fully disclosed herein, the interrelationship of the various faces of the eight cubes result in a systematic numbering scheme for the entire set of 8 cubes and for the dodecahedron composed thereof.

One or more die formed in this manner can be used as a general dice game, as well as for card games, and other games. In one respect, the prior art U.S. Pat. No. 1,517,113 mentioned above, is similar to the present invention in that it shows a die structure having a rhombic dodecahedron configuration. However, this prior teaching proposes that the die be used as a single die for a baseball game in which the various die faces represent typical baseball plays. Furthermore, because this prior art patent does not derive the rhombic dodecahedron shape from a four dimensional system, it does not disclose the face numbering formula characterizing the present invention as more fully disclosed herein.

Thus, it is an object of the present invention to provide a new game using dice formed by degenerating or reducing a four dimensional concept into a three dimensional multi-faced solid object or die.

Also, it is an object of the present invention to provide an educational toy demonstrative of the inter-relationship between a multi-dimensional concept and the representation thereof as three dimensional objects, such as one or more die.

It is another object of the present invention to derive such a three dimensional multi-faced die having its faces numbered in accordance with self-consistent numbering systems based on the four dimensional concept.

A still further object of the present invention is to provide one or more die derived in the foregoing manner to define a solid object having the shape of a dodecahedron defining 12 rhombic shaped faces. These 12 faces are numbered by a numbering system for the eight individual and interrelated cubes composing the dodecahedron. In the resulting dodecahedron die, only 12 external faces show, however the numbers or values of these 12 faces have been determined by the remaining and hidden faces of the above mentioned eight cubes.

It is another object of the invention to provide such a 12 faced die, in which additional indicia means are applied to the die faces also in accordance with the eight cubes comprising the 12 faced die. For example, the indicia means may be a color code in which different faces are marked with a different color. One or more die formed in this manner may be employed in a game in which both the face numbers and set colors are used as probability determining events.

This invention also has an object, the provision of a game board, such as a crap board, in which the standard betting zones of the crap board are provided with sub-zones to permit a player to place a bet dependent on both or either of the number or color of the "up" faces of the rolled dice.

An additional object of the present invention is to provide a dice game in which the 8 individual cubes which comprise the degenerate four dimensional object are broken out from the four dimensional object and used as a game composed of 8 cubic dice. The six faces of each of these eight dice are provided with values, such as card values including a designated suit, in which the values and suits are determined from the numbering system characterizing this invention. In other words, the face values of the eight dice are all interrelated in accordance with the numbering system of the present invention which in turn is derived from the four dimensional concept represented in three dimensional form.

These and further objects and various advantages of the pseudo four dimensional dice and game according to the present invention will become apparent to those skilled in the art from a consideration of the following detailed description of an exemplary embodiment thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference will be made to the appended sheets of drawings in which:

FIG. 1 is a perspective view of a pair of degenerate, pseudo four dimensional dice formed in accordance with the present invention.

FIG. 2 is an isometric diagrammatic representation of the generation of a pseudo four dimensional die from a pair of intersecting three dimensional cubes.

FIG. 3 is another isometric diagrammatic representation of the generation of the dice in FIG. 1 in which the corners of the three dimensional cubes shown in FIG. 2 are joined to form a solid figure which is shown to comprise 8 separate cubes shown surrounding the central figure.

FIG. 4 is a table setting forth the self-consistent numbering system by which the six faces of each of the eight cubes forming the 12 face die and the 12 faces of the pseudo four dimensional die are numbered in accordance with the present invention.

FIG. 5 is a front elevational view of a 12 sided die in which the faces thereof have been provided with number and color indicia means in accordance with the present invention.

FIG. 6 is a top plan view of the die shown in FIG. 5 as viewed from VI--VI therein.

FIG. 7 is a rear plan view of the die as seen from VII--VII in FIG. 6.

FIG. 8 is a bottom view of the die as seen from VIII--VIII in FIG. 7.

FIG. 9 is a front elevation view of a die, similar to FIGS. 5-8 but showing a different arrangement of the numbers still in accordance with the self-consistent numbering formula of the present invention.

FIG. 10 is a top plan view of the die of FIG. 9 taken from X--X therein.

FIG. 11 is a plan view of a conventional game board for the dice game sometimes called craps, showing the various betting zones available to the players.

FIG. 12 is a view of a game board similar to FIG. 11, but here modified in accordance with the present invention to provide betting sub-zones in accordance with the various indicia means provided on the dice shown in FIGS. 1 and 5 through 10.

DESCRIPTION:

With reference to FIG. 1 one embodiment of the game of the present invention includes one or more die 9 and 10 having a shape resulting from the degeneration of a four dimensional conceptual system into a three dimensional representation. In this instance and as shown in FIG. 1, the three dimensional representation of each die is in the form of a rhombic dodecahedron, or stated differently, a 12 sided three dimensional object having faces in the shape of rhombi.

Because these dice are derived in this manner, they may be referred to as pseudo four dimensional dice.

The dice in accordance with the present invention may also be termed degenerate or pseudo "tesseracts". The term tesseract is the mathematical name of a four dimensional hypercube derived from the Ionic tesseres meaning four, plus "aktis" meaning ray. The coordinate system comprising the four dimensional system includes four vectors (rays), each perpendicular to the other three vectors as will be explained more fully herein. Thus, the 12 sided die is a three dimensional object representative of the four dimensional tesseract, and thus the dice may also be called pseudo tesseracts. Although 12 sided die have been known for some time, the manner in which a particular type of 12 sided die, namely a rhomic dodecahedral die, may be employed to represent a four dimensional concept and the self-consistent numbering formula used in marking and distinguishing the 12 die faces in accordance with the present invention are unique. To understand the derivation of these dice from tesseracts, it is helpful to consider the following background in geometrical representations.

In mathematics a point has no dimensions. A one dimensional line may be generated by translating a point through a length L so that it consists of two points or vertices and a single line or edge.

If this line is translated through a distance L in a direction perpendicular to the single edge of the line, the figure generated is a two dimensional square with four vertices, four edges and one face.

From this square a cube may be generated by translating the two dimensional square through a distance L perpendicular to the two dimensional plane to form a three dimensional co-ordinate system. The cube has eight vertices, 12 edges, six faces and one volume.

It is thus observed that it is possible to draw a point, a line and a square since the page is two dimensional and since these representations are equal to or less than two dimensional objects. The cube can be represented in the form of a two dimensional "picture". However, this is only a representation and its true shape is one of a three dimensional object.

Extending this analysis one further step into a fourth dimension, it is possible to conceptually construct a four dimensional system or tesseract hypercube by imagining a cube at a time to and the infinite sequence of cubes ending at a time t1. In other words, the fourth dimension is realized by using the dimension of time rather than a spatial dimension. Stated differently, the fourth dimension is obtained by translating the three dimensional solid object along a time axis for a length L, such that L = c (t1 - to), where c is the velocity of light.

This four dimensional hypercube can be constructed in pictorial form in three dimensions in the same sense that a three dimensional solid object can be pictorially represented in a two dimensional picture. Thus, the tesseract or four dimensional hypercube is bounded by 16 vertices, 32 edges, 24 faces and eight cubes or volumes. This is compared with a cube in which the volume is bounded by eight vertices, 12 edges and six faces; and with a two dimensional square which has four vertices, four edges, one face and no volume.

By representing the four dimensional hypercube or tesseract in three dimensions and then taking a two dimensional picture of the three dimensional representation, the tesseract or four dimensional hypercube may be depicted as shown in FIGS. 2 and 3. The result may be referred to as a pseudo four dimensional hypercube or pseudo tesseract.

The foregoing is graphically illustrated in FIG. 2, by a cube 11 shown intersecting with a cube 12. The edges of each of cubes 11 and 12 are drawn along the three spatial dimensions or co-ordinates represented as x, y, z. The fourth dimension is shown as the translation of cube 12 along an axis w which is conceptually perpendicular to the two dimensional picture representation of cube 11 in the two dimensional plane of the drawing. This conceptually represents the four dimensional hypercube or tesseract in which the fourth dimension is a perpendicular translation of the two dimensional picture of the three dimensional cube 11. The cube 12 is the result of translating the two dimensional picture of cube 11 along the time or fourth dimensional axis w.

The representation in FIG. 2 may be developed into a two dimensional picture of a three dimensional tesseract as shown in FIG. 3 by diagonally connecting the vertices or corners of the intersecting cubes 11 and 12 as illustrated. By connecting these vertices and then dissecting the resulting picture into its component parts, it is possible to demonstrate the presence of eight separate identifiable cubes or volumes which comprise the tesseract. These eight separate but interrelated cubes are shown as cubes 21, 22, 23, 24, 25, 26, 27, and 28 surrounding the central tesseract.

With reference to FIG. 4, each of these eight cubes comprising the tesseract and each of the six faces of each such separate cube may be uniquely identified by the indicated positive (+), negative (-) and parallel (l) notation. Basically, the edges of the cubes define four sets of eight parallel lines comprising the tesseract figure. If the direction to the right along each of the tesseract edges is called positive (+), the direction to the left along each edge called negative (-), and the set of lines parallel to a given dimension as (l), then each of the six faces of each of the eight cubes 21 through 28 may be uniquely identified as shown in the table of FIG. 4. Thus the cube 21 is comprised of a face parallel to the y and z axis (denoted as y = l and z = l) and to the right of the x axis (denoted x = +) and to the left of the w axis (denoted w = -). Similarly the other faces of cube 21 are identified by the plus sign (+), minus sign (-), and parallel (l) notation under each of the x, y, z, and w axis.

To create a self-consistent numbering system for the eight cubes comprising the tesseract, the numbering system for a standard or regular six sided cubic die may be used as a starting point. Starting with any one of the eight cubes comprising the tesseract of FIG. 3, each of the six faces of such cube is designated as a number or indicia 1 through 6 with the numbers on opposite parallel faces adding to seven in a manner similar to the numbering of a regular six sided die. Thus as shown for cube 21, the 2, 3 and 6 value faces are shown. This automatically fixes the remaining face values as 5, 4 and 1 respectively located on the opposite parallel faces of this die. The numbers for the 5, 4 and 1 faces for die 21 have been deleted for clarity.

Having allocated values for the six faces of one tesseract cube, the six faces on each of the remaining seven cubes can be similarly determined. This determination however of the remaining faces must take into account the fundamental characteristics of the tesseract and the eight volumes or cubes of which it is comprised.

These characteristics are based on (1) The intersection of tesseract cubes 21 through 28 in common faces, i.e., each face of the geometrical structure is shared by two of the tesseract cubes. Accordingly, this face shared by two cubes must have the same value or indicia, unless the face is internally of the three dimensional pseudo tesseract, in which case an arbitrary value may be assigned as the face is hidden from view. (2) There must be some rule relating opposite faces of each of the six sided tesseract cubes. For example, the rule derived from regular dice that opposite faces of each cube must add up to seven may be used as in the case of the presently disclosed embodiment.

Thus with reference to FIG. 4, the six faces of cube number 25 are designated as indicated under the column identified as face value. Each of these faces of cube 25 are shared with one other cube, and such shared face must of course carry the same face value. Accordingly, the face in FIG. 4 identified as x = l, y = -, z = l, and w = +, has been given the value l on cube 25, and this same face is shared with the last face of cube 28 is indicated by the interconnecting line. Accordingly, this face of cube 28 also has the face value l. Similarly, the face value 6 of cube 25 is shared with the last face of cube 24; face value number 2 of cube 25 is shared with the last face of cube number 22; the face having a value 5 of cube 25 is shared with the last face of cube number 26; the face value 3 is shared with the last face of cube number 27; and face value 4 is shared with the last face of cube number 23. It is observed that the shared faces have the same sign notation even though they are associated with different numbered cubes. Thus the face value 1 of cube 25 may be located by finding the same sign notation, which as indicated above re-appears as the last face of cube number 28. Thus in completing the numbering scheme, there is a limitation on the freedom of choice for the face values. For example, once the face values for cube number 25 have been assigned, this necessarily determines the values of at least six and as many as 18 other faces of the 48 faces defined by the 8 cubes.

It is noted with respect to FIG. 3, that when the 8 tesseract cubes are consolidated into the three dimensional pseudo tesseract, that only certain of the 48 cube faces appear externally of the figure. That is, the pseudo tesseract is a 12 sided or dodecahedral object in which the 12 external faces correspond to certain of the 48 faces defined by the eight six-sided tesseract cubes. In FIG. 4, the external faces can be readily identified from the face notation, and these are shown in FIG. 4 by the square box around certain face values. Similarly, the internal or hidden faces can be identified and these are shown in FIG. 4 by the circled face values. It is observed that the external face values are to a certain extent controlled by the internal face values of the complete set of eight tesseract cubes. As a result of this self-consistent numbering system, the 12 external faces of the pseudo tesseract carry two sets of face values numbers 1 through 6.

Furthermore, each of the eight tesseract cubes contributes three external faces and three internal faces to the degenerate tesseract figure. Thus in FIG. 3, cube number 25 contributes external faces 1, 5, and 3; cube number 23 contributes exterior faces 2, 6, and 3, etc.

A further characteristic of this system is that the cubes 21 through 28 form mirror image pairs of one another. Thus cubes 21 and 25 are mirror images of one another; cubes 22 and 26 are mirror images; cubes 23 and 27 are mirror images; and cubes 24 and 28 are mirror images.

If an attempt is made to number the 8 cubes in accordance with traditional practice associated with regular die, then only certain of the cubes will exhibit the standard counter-clockwise numbering convention. In particular, this convention provides that if a regular die is viewed alone one of its diagonals, that the faces designated one, two and three will be adjacent one another and will exhibit a counter-clockwise movement when going from face number 1 to number 2 to number 3.

With respect to the present invention however, only certain of the eight tesseract cubes can have the standard counter-clockwise numbering convention, because of the characteristic mirror image pair formation. Accordingly, in the present disclosure, cubes numbers 22, 23, 24, and 25 have their six faces numbered in accordance with the standard counter-clockwise convention, whereas cubes numbers 26, 27, 28 and 21 are mirror images thereof requiring a clock-wise convention in their face values.

Furthermore, the opposite cube pairs may be identified as a plus or minus cube of an associated pair, depending upon which end (+ or -) of the fourth axis, i.e. the axis not used in forming the cube, that the designated cube is found. Thus, the cubes 21 and 25 are opposite pairs having negative and positive values respectively as indicated in FIG. 3. Similarly, the cubes 22 and 26, 23 and 27, 24 and 28 are opposite plus and minus pairs as indicated in FIG. 3.

It takes six face assignments on three tesseract cubes of the set of eight here disclosed to consistently number all 48 faces of the tesseract. This may be demonstrated as follows. Using the rule that opposite faces of each individual cube must add to seven, and following the chart in FIG. 4 to indicate common or shared faces of the tesseract, the following faces values may be traced. Given the face value 1 on cube number 25, this automatically and necessarily designates the value of the last face of cube 28. Since opposite faces of each cube add to seven, this determines the face value 6 for the second to last face of cube 28, observe sign notation for opposite faces, which in turn establishes the value of the first face of cube number 25. The opposite face of cube 21 is thus designated 1, here the second face of cube 21, which inturn sets the valve for the same face which is shared with cube number 24. The opposite face of cube 24, here the last face thereof, is thus 6 in accordance with the addition to seven rule. The face designated 6 of cube 24 is shared with the second face of cube 25, which is thereby assigned the value 6. This forms a complete set of eight faces one 4 cubes consisting of two pairs of mirror image cubes. Accordingly, given one number on a particular cube face, this automatically sets the numbers on eight faces of four cubes. The numbers 1, 2 and 3 can be assigned any of three faces of the first cube namely, cube number 25, which automatically sets, as shown above, the other three faces of the selected cube and 18 other faces on the remaining seven cubes. Examining cube number 23, only two faces have assigned values 3 and 4. Therefore the numbers 1 and 2 may be assigned to two of the remaining faces which automatically sets 16 more faces, or a total of 40 faces thusfar. One remaining pair of faces of cube number 28 can be assigned the remaining number pair 3 and 4 so that all 48 faces have now been assigned numbers such that each cube of the tesseract has all numbers 1 through 6 assigned. A picture of a true tesseract together with its self-consistent numbering system as described above may be formed into a three dimensional picture-like representational solid by connecting opposite diagonal corners of the cubes 11 and 12 to define the outer twelve boundary surfaces of the solid. The remaining twelve faces are interior to the three dimensional representation. The faces for such a pseudo tesseract or pseudo four dimensional die will carry the face values as developed in the foregoing self-consistent numbering system.

For the pseudo tesseract shown by the central three dimensional solid in FIG. 3, a simplified or short-hand notation of the numbering system can be devised as follows:

Rule A: The face up and the face down of the pseudo tesseract will have the same value, or stated differently, opposite parallel faces are provided with the same indicia.

Rule B: The sum of the represented number or value on any given face of the pseudo tesseract plus the number or value on another face connected thereto only by a vertex equals the sum of seven.

Rule C: The sum of the value or number on any given face plus the number or value on another face connected thereto by an edge does not equal seven. (Note: Rule C is redundant and follows as a necessary result of the application of Rules A and B if the six faces of each of the eight tesseract cubes 21 through 28 are assigned all six values of numbers 1 through 6.)

It is observed that the Rules A through C above are merely the result of the self consistent numbering system taught in connection with FIGS. 2, 3 and 4 in the above disclosure.

The pseudo four dimensional die or tesseract resulting from FIGS. 2 and 3 is the three dimensional representation or picture of the tesseract from some arbitrary direction with respect to the four axis x, y, z and w. The two cube vertices interior to the solid can have almost any position, constrained only by the distortions which maintain constant length for all cube edges. Although any such solid three dimensional object formed and numbered in this manner may be used as the pseudo four dimensional die, a particular configuration is preferred. This preferred configuration is the equivalent unique degenerate pseudo four dimensional tesseract derived by joining the two interior cube vertices, of cubes 11 and 12 in FIGS. 2 and 3, to form a dodecahedron having rhombic faces. A rhombic dodecahedron or degenerate pseudo four dimensional configuration is preferred because of its symmetry, permitting a die to be formed which will have an equal probability of coming to rest on any one of its 12 faces. Although the three dimensional object formed merely by joining diagonal corners of the cubes 11 and 12 as shown in FIG. 3 has 12 sides like the rhombic faced dodecahedron, it does not have the three-fold and four-fold symmetry axes of a rhombic-faced dodecahedron and thus there is an un-equal probability that its various sides will land face up after a dice roll. The preferred geometrical configuration for each die may be formed by starting with the 12 sided figure as shown in FIG. 3 and, keeping all edges equal in length, distorting its various faces until it degenerates into the rhombic-faced dodecahedron. This leaves the 12 faces of the dodecahedron with the same numbering system discussed above in connection with FIGS. 3 and 4.

The resulting die may be used in a game either by itself or with one or more additional die having the same or similar face markings. In FIG. 1 a pair of rhombic dodecahedral die are illustrated with their faces appropriately numbered in accordance with the foregoing system. With reference to FIGS. 5 through 10 an enlarged view of one of these die is illustrated. As indicated, each of the 12 die faces is divided into two separate sets of six faces each corresponding to the numbering system derived from the true four dimensional hypercube or tesseract die. FIGS. 5 through 8 depict separate plan views of the die as it is rotated 90° in each view. Accordingly, all 12 faces of the die are shown collectively in the FIGS. 5 through 8.

Each set of six faces is provided with number indicia means for representing the numbers 1 through 6. Thus, for example, in FIG. 5 the numbers 1, 2, 5, 3 and 4 are represented by indicia means, such as dimples, pips or other suitable means, on the die faces 31, 32, 33, 34 and 35 respectively. With reference to the remaining faces of the die shown in FIGS. 6, 7, and 8, it will be observed that the 12 faces are numbered in accordance with the above defined rules A, B and C as follows. The number indicia means on opposite parallel faces, such as faces 31 and 36 shown in FIGS. 5 and 7 respectively represent the same number, in this instance the number 1.

Secondly, the sum of the represented number on any given face, such as face 31, and the number on another face which is perpendicular to the first chosen face equals the total of seven. Thus, the number 1 on face 31 plus the value 6 represented on face 37 in FIG. 6 equals the total of seven. Similarly, the value 1 on face 36 plus the value 6 on face 38 in FIG. 8 again total seven.

Finally, the sum of the represented number on any given face, such as the number 1 on face 31 in FIG. 5, plus the number on another face, such as any one of faces 32, 33, 34 and 35 does not equal the total of seven. Accordingly, the value 1 on face 31 plus the value 2 on face 32; the value 1 on face 31 plus the value 5 on face 33; the value 1 on face 31 plus the value 3 on face 34; and the value 1 on face 31 plus the value 4 on face 35 do not equal seven. Note that this may be a necessary result of the application of the first and second rules above in accordance with the discussion hereinabove.

Although one or more die formed and numbered in this manner may be used in accordance with the game of the present invention, a preferred embodiment of this invention provides further indicia means on the various die faces.

Such additional or further indicia means may be provided in the form of differently colored faces. These different colors may be superimposed or otherwise used concurrently with the number values for the faces of each die.

In accordance with the foregoing scheme for numbering the die faces, the first rule to be observed is that all shared faces of the tesseract cubes must be identical both in color and numerical value. Secondly, no more than three faces of each cube shall have the same color. If these rules are followed all exposed faces of the pseudo tesseract will have identical color for identical face number values. If however the odd numbers are provided with one color such as red, as in roulette, then the even numbers can be provided with a different color, such as black. Alternatively, exterior faces 1, 2, and 3 may be designated red and the remaining faces 4, 5, and 6 black.

A still further and preferred alternative is to relax the rules requiring that shared faces have the same color value, and permit interior faces, that is those faces hidden inside the degenerate tesseract to have an indeterminate color. By relaxing the rule that a shared face of any two of the eight tesseract cubes may be different, so that opposite tesseract faces can also be different.

Accordingly, in accordance with the presently preferred embodiment, opposite or parallel faces of the degenerate tesseract die shown in FIGS. 5 through 10 are provided with different color indicia means, such as the colors red and black. Accordingly, as opposite faces have the same numerical value, this differentiates between the two faces carrying the same numerical value. Thus each 12 sided die in FIGS. 5 through 10 will have a red 2 and a black 2, a red 3 and a black 3, etc.

In this instance, in addition to the red and black color indicia means for the two sets of face values 2-6, a third color or indicia means is provided for the faces 31 and 36 carrying the value 1 as discussed below.

By utilizing the color or set indicia means for differentiating between the two sets of six faces each on a die, it is possible to use the die or dice in a variety of games. For example, the colored faces may be used in a game similar to Roulette, in which a number, such as 1 through 6 may be called out in combination with a color such as black or red with the probability of the event being determined by both the number indicia means and the set or color indicia means. If, for example, a red-2 is called out, that means the player must roll the die such that the face 32 is up-right at the end of the dice roll. Furthermore, a pair of dice numbered and colored in this manner may be used to advantage in a game which depends both on the combined or total number that is rolled and also on the combined color of the roll. Thus the odds or probability of rolling two black fours will be different and less probable than rolling a pair of fours, one black and the other red.

As a further example, given a total point count, such as nine, it is possible to roll this point by any of the following combinations: black-5 plus black-4, black-5 plus red-4, red-5 plus black-4, red-5 plus red-4; black-6 plus black-3, black-6 plus red-3, red-6 plus black-3, red-6 plus red-3; and black-7 plus black-2, black-7 plus red-2, red-7 plus black-2, red-7 plus red-2. Thus the possible variations on rolling a particular "point" have been substantially increased. The combined probability of rolling any of these various numbers and colors can be determined and incorporated into the dice game.

Furthermore, in accordance with the present invention, these dice are preferably provided with a still further indicia means, in this instance being provided by a third color distinguishing one set of faces of like value from all of the remaining faces. This is for the purpose of slightly alterating the odds so as to favor one player or the other as desired. In the present embodiment, as illustrated in FIGS. 5 through 8, the faces 31 and 36 carrying the number indicia means representing one are colored green or any other third color different from the pair of colors, here red and black, differentiating the two sets of six faces. Thus in a game depending on the color that is rolled, the green faces 31 and 36 function as the zero and double zero in Roulette altering the odds in favor of the house. It is observed that when a one spot comes up, it pairs with black, red or green. Thus introducing an additional variable in the color combination discussed above and appearing only in the event a value 1 is rolled on one of the die. A roll resulting in a pair of green die faces corresponds to snake-eyes, i.e., one and one.

The degenerate tesseract or 12 sided die shown in FIGS. 5 through 8 has its number face values arranged according to one of several possible arrangements satisfying the rules above. The die shown in FIGS. 9 and 10 illustrates another arrangement of the face values. Not all 12 faces of the die are shown in FIGS. 9 and 10 as the values for the faces which are shown determine the values for the remaining faces. Thus, based on the illustration in FIGS. 9 and 10 it is possible to add the remaining face values by merely applying rule A which requires that opposite parallel faces carry the same number value.

In one embodiment of the dice game in accordance with the present invention, a game board is provided for use in combination with the degenerate tesseract dice. The game board may be of a type having betting zones for allowing players to place bets with or against the occurrence of certain numbers, colors or combinations of both. The numbers and colors provided by the number and set indicia means serve as the probability determining events, in a manner similar to the use of standard dice. The game board may be based on a standard game board, such as shown in FIG. 11 for use in playing the dice game, sometimes called craps. FIG. 11 illustrates one side of a standard crap board having the usual betting zones for allowing players to place their desired bets. Thus, there is provided a PASS LINE betting zone 41, a FIELD betting zone 42, a COME betting zone 43 and a DON'T PASS/DON'T COME betting zone 44. These are the basic betting zones of the conventional crap game. In addition, the game board as indicated in FIG. 11, may provide point designations for the dice values 4, 5, 6, 8, 9 and 10 as indicated at 46 to provide a place where the players point may be indicated by placing his bet either in front of or behind the respective numbers to indicate a PASS or DON'T PASS bet respectively. Similarly, the box as indicated at 47 over-lying the boxes for the values 4, 5, 6, 8, 9 and 10 provide bet indicating zones for the COME and DON'T COME bets. Particular number bets together with the odds therefore may be designated in a zonee 48 called in this instance NUMBER BETS. Similarly, bets for ANY CRAPS for betting on any roll that 2, 3 or 12 will occur, is provided in a zone 49 as shown in FIG. 11.

In accordance with the present invention this conventional gaming board is reconfigured to provide for the additional combinations of face colors and face values of the degenerate four dimensional hypercube dice shaped and marked as described above. Thus, in FIG. 12, the conventional betting zones of PASS LINE 41', FIELD 42', COME 43', and DON'T PASS/DON'T COME 44' are provided with zone indicia means corresponding to both the number indicia means and set indicia means on the die faces. This permits the player to place a bet dependent on the occurrence of a particular number or total of numbers on a pair of dice and/or dependent on the color or combinations of colors represented by the set indicia means differentiating between the two sets of values 1 through 6 on each die.

Accordingly, the PASS LINE betting zone 41' is divided into betting sub-zones 51 and 52 representing and corresponding to the different colors, such as black and red in this instance, of the set indicia means on each die. An additional sub-zone 53 representing and corresponding to the additional color indicia means, in this instance green, may also be provided. To allow a player to bet on the number value of the dice only, a plain subzone 54 may be provided as part of the over-all PASS LINE zone 41' as indicated.

Similarly, each of the FIELD, COME, DON'T PASS/DON'T COME zones 42', 43' and 44' are marked with color sub-zones corresponding to the color indicia means on the dice. Thus, FIELD betting zone 42' is provided with a black sub-zone 56, a red sub-zone 57, a green sub-zone 58 and a plain sub-zone 59 as shown. COME betting zone 43' has a black sub-zone 61, a red sub-zone 62, a green sub-zone 63 and a plain sub-zone 64. Black, red, green and plain sub-zones 66, 67, 68 and 69 are similarly provided for the DON'T PASS/DON'T COME betting zone 44'. Corresponding color sub-zones are formed in the bet indicating boxes 46' and 47' as shown.

Preferably, the game board color sub-zones are laid out to be adjacent one another so that a player can straddle the various sub-zones with his betting token. This enables the player to select the color individually or in combinations. As two dice are used in the game, the color possibilities upon any roll are as follows: both black, both red, both green, one black and one red, one black and one green, and one red and one green. It is observed that all of the color sub-zones, such as the PASS LINE sub-zones 51, 52 and 53 are disposed so that each color borders the other two colors permitting a token to straddle a border between any two color sub-zones. Thus, for example, a player may place a token bet with the PASS LINE zone 41' so as to straddle the border between black and red color sub-zones 52 and 51. This means the player is betting that he will roll a 7 or 11 with a color split of red and black. Thus, a roll of a black 6 and a red 1 will win on this particular bet. Similarly, the bet can be places so as to straddle zones 51 and 53 or zones 52 and 53 requiring a combination of red and green and black and green respectively.

It will be appreciated that the probability of winning on a bet in which the color of the die faces is designated, is less than if the plain face value of the dice are played by a bet within plain sub-zone 54.

An additional betting region may be provided as indicated at 71 on the game board of FIG. 12 just below the ANY CRAPS betting zone 49'. Region 71 is a special betting zone for placing bets dependent only on the occurrence of the colors of the set indicia means on the dice. In other words, a player can place a bet based on the colors only while other players are rolling the dice for either the numbers or numbers and colors in combination. Thus in zone 71 a plurality of color sub-zones 72, 73 and 74 are provided corresponding to the colors of black, red and green in this instance. Again, the color sub-zone 72, 73 and 74 are placed in adjacency so as to allow a player to straddle any pair of colors corresponding to the pair of dice to be rolled. In the betting zone 48' called in this instance NUMBER BETS, the individual betting combinations are set forth together with the odds or probability of rolling the particular combination. These number bets may include not only the numbers, such as snake-eyes, double twos', double threes', etc., but also the color combination of black-black, red-black, green-black, green-green, etc. The following schedule is illustrative of the various die value and color combinations possible together with the random odds. These odds may be used as a basis for token pay-offs. For example, a pay-off might be 30:1 for random odds of 35:1.

PROBABILITY TABLE __________________________________________________________________________ Point Combinations Random Odds __________________________________________________________________________ BR RR/BB GR/GB GG BR RR/BB GR/GB GG 2 -- -- -- 4 -- -- -- 35:1 3 -- -- 4 -- -- -- 35:1 -- 4 2 1 4 -- 71:1 143:1 35:1 -- 5 4 2 4 -- 35:1 71:1 35:1 -- 6 6 3 4 -- 23:1 47:1 35:1 -- 7 8 4 4 -- 17:1 35:1 35:1 -- 8 10 5 -- -- 67:5 139:5 -- -- 9 8 4 -- -- 17:1 35:1 -- -- 10 6 3 -- -- 23:1 47:1 -- -- 11 4 2 -- -- 35:1 71:1 -- -- 12 2 1 -- -- 71:1 143:1 -- -- 7-11 12 6 4 -- 11:1 23:1 35:1 -- 2,3,12 2 1 4 4 71:1 143:1 35:1 35:1 Color Only 50 25 20 4 47:25 119:25 31:5 35:1 __________________________________________________________________________

The rhombic faced dodecahedron dice as shown in FIGS. 1 and 5 through 10 may be fabricated from known materials and in accordance with existing techniques. For example, the dice can be either opaque or transparent. To keep the dice honest, a transparent material may be used. In a transparent die, the number indicia means may be provided by pips or dimples, as suggested in FIGS. 5 through 10, and the pips may be colored selectively on each face to serve as the color indicia means disclosed herein.

A further preferred embodiment of the invention provides for the creation of card dice for playing a card game such as poker. Although poker dice are in general known, the present invention provides a particular and unique form of poker dice derived from the tesseract discussed above in connection with FIGS. 2, 3 and 4. One form of the poker dice in accordance with the present invention involves the use of the eight tesseract cubes 21 through 28 of FIGS. 3 and 4, in which the six faces of each cube are assigned card values including the face value such as Ace, King, Queen, Jack, 10, 9, 8 etc., and the suit such as Spade, Heart, Diamond, Club. As only 48 faces are available, six faces on each of the 8 tesseract cubes, a 48 card deck is represented, in which the four two's are omitted.

The game may be played by taking the eight tesseract cubes, having their respective faces marked with the card values in accordance with the present invention as discussed herein, and rolling the eight cubes as dice such that the up-face values represent the players card hand. To provide the uniformity of probability that any given card will show up in a players hand, it is necessary to distribute the 48 card values and suits evenly throughout the 48 available faces of the tesseract cubes. This can be accomplished in accordance with the present invention as follows.

Reference is made to the translation table set forth

*TRANSLATION TABLE ______________________________________ Code Nominal Translated Code Nominal Translated No. Face Face No. Face Face Value Value Value Value ______________________________________ 1 1 6 6 6 7 1 12 25 2 11 21 5 8 5 5 2 2 3 10 4 9 4 4 3 3 2 11 5 5 5 8 2 2 23 1 1 27 6 6 6 7 1 12 3 3 4 9 4 4 3 10 2 2 5 5 5 8 2 11 24 3 3 28 4 4 4 9 3 10 1 12 6 6 6 7 1 1 1 12 6 6 6 7 1 1 26 3 10 22 4 4 4 9 3 3 2 2 5 8 5 5 2 11 ______________________________________ *Translated Value determined by applying rule that opposite faces of tesseract add to 13 and adjacent pair number differences are greater than or equal to 2.

herein, in which the nominal face values of 1 through 6 for each of cubes 21 through 28 as set forth in FIG. 4 are translated to face values 1 through 12. This translation is effected by applying the rule that opposite faces of the degenerate tesseract in FIG. 3 add to 13 and that adjacent pair number differences are greater than or equal to 2 as a result the individual face values for cubes 21 through 28 previously having only the nominal values 1 through 6, now have translated face values of 1 through 12.

Having accomplished this translation, it is possible to substitute the translated face values of 1 through 12 with the face value of a deck of cards, such as card values 3 through 10, Jack, Queen, King and Ace.

If this is done in the manner disclosed herein, a set of card dice may be developed in which a variety of card games including poker may be played. The card dice game may take several forms in accordance with the present invention, one of which involves the use of the eight cubes forming the tesseract as a combined card hand in the following manner.

In one form, the die scheme in the present invention contemplates the use of the eight tesseract cubes of FIG. 3, having their various faces representing the face values and suits of a deck of cards. For example, the 48 faces available with the eight tesseract cubes, may be used to represent a 48 card deck, with the four cards having the face value of two omitted. These eight cubes or dice may be rolled together as a unit, and the resulting up faces, showing the face value in suit of the card, will represent the hand of the player. From these eight card faces, a smaller number, such as five cards may be selected and played as in the case of five card draw poker.

To number the 48 faces of the set of eight dice, it is necessary to utilize the self-consistent numbering system disclosed above in connection with FIG. 4, and the translation table discussed above. With these tables, a poker dice table may be developed as shown herein, in which the 48 faces of the eight available tesseract cubes are assigned card values and suits.

With reference to the poker dice table set forth herein, the top row shows the number of the cube, corresponding to the reference numerals 21 through 28 of FIGS. 3 and 4. The left-hand column shows the translated face value of 1 through 12 derived from the translation table above. The right-hand column shows the assigned card value of 3, 4, 5, 6, 7, 8, 9, 10 Jack, Queen, King and Ace. These assigned card values correspond directly with the translated face values 1 through 12. The row of cube numbers and the column of poker dice values form a matrix in which the circled letter S, H, D and C correspond to assigned suits of Spade, Heart, Diamond and Club. This is accomplished in the following manner. For the card value 3, it is necessary to have four different suits of Spade, Heart, Diamond and Club from the available 48 cube faces. Thus starting with cube 21 from the translation table it is observed that the translated value 1 does not appear on cube 21, therefore the corresponding poker dice value of three is not assigned to any face on cube 21.

Next, observing that cube 22 does have a translated face value of 1, this may be arbitrarily assigned the club suit or C, indicating that cube number 22 carries a face representing the three of clubs. Surprisingly, it has been found that each translated face value and corresponding poker dice value occurs four times among the 8 available cubes, such that each occurrence can be designated by a different suit. In this manner the matrix shown in the poker dice table is completed to form a 48 card set of poker dice, consisting of 8 cubes or eight six sided dice representing the indicated card values and suits.

To allow the formation of such hands as "four of a kind", and "flushes", it is necessary to arrange the suit

*POKER DICE TABLE __________________________________________________________________________ SUIT & VALUE FOR EACH FACE Tesseract-Cube Translated Poker Face Cube No. Dice Value Value 21 22 23 24 25 26 27 28 __________________________________________________________________________ 1 C H S D 3 2 C S H D 4 3 D H S C 5 4 S D H C 6 5 D C H S 7 6 S D C H 8 7 H D C S 9 8 D C S H 10 9 H C D S J 10 H C D S Q 11 D C S H K 12 S D H C A S -- SPADE H -- HEART D -- DIAMOND C -- CLUB __________________________________________________________________________ *A self consistent set of face numbering and suit distribution which depends on tesseract cubes.

designations in the illustrated poker dice table such that each suit is separated by at least four values or numbers. Thus, cube number 21, for example, does not carry both the four of spades and the five of spades or any other spade within four cards of the four of spades. Similarly, for cube number 22, the three of clubs is separated by four faces before the club suit reappears, in this instance at the eight of clubs. By designating adjacent card values of the same suit on different cubes, it is therefore possible to roll straight flushes and royal flushes.

While only a limited number of embodiments of the present invention have been disclosed herein, it will be readily apparent to persons skilled in the art that numerous changes and modifications may be made thereto without departing from the spirit of the invention.