US3788645A

US3788645A - Mathematical cube puzzle

Info

Publication number: US3788645A
Application number: US00258872A
Authority: US
Inventors: S Nelson
Original assignee: Individual
Current assignee: Individual
Priority date: 1972-06-01
Filing date: 1972-06-01
Publication date: 1974-01-29
Anticipated expiration: 1991-01-29

Abstract

A puzzle made up of four separate cubes which are arrangeable into a plurality of differing solutions is described. Each edge of each cube has one of a set of three colors associated therewith, and the resulting arrangement or combination of colors around the edges on each cube face is different from that on every other cube face, and the colors of all common edges of adjacent faces on each individual cube match one another. The object of the puzzle is to arrange the various cubes relative to one another so that the colors associated with all exposed adjacent playing edges of different cubes match one another, e.g., are the same. Various solutions of differing degrees of difficulty are described satisfying this criteria.

Description

United States Patent [191 Nelson 0 MATHEMATICAL CUBE PUZZLE [21] Appl. N0.: 258,872

[52] US. Cl. 273/156, 273/137 D [51] Int. Cl. A63f 9/08 [58] Field of Search 273/157 R, 156, 137 D [56] References Cited UNITED STATES PATENTS 12/1970 Williams et al. 273/157 R 2/1865 Harold 273/157 R UX OTHER PUBLICATIONS New Mathematical Pastimes by P. A. MacMahon, published 1921 by Cambridge University press.

[ Jan. 29, 1974 Primary Examiner-Anton O. Oechsle Attorney, Agent, or FirmMoore, Zimmerman &

Dubb

[ ABSIRACT A puzzle made up of four separate cubes which are arrangeable into a plurality of differing solutions is described. Each edge of each cube has one of a set of three colors associated therewith, and the resulting arrangement or combination of colors around the edges on each cube face is different from that on every other cube face, and the colors of all common edges of adjacent faces on each individual cube match one another. The object of the puzzle is to arrange the various cubes relative to one another so that the colors associated with all exposed adjacent playing edges of different cubes match one another, e.g., are the same. Various solutions of differing degrees of difficulty are described satisfying this criteria.

5 Claims, 3 Drawing Figures MATHEMATICAL CUBE PUZZLE BACKGROUND OF THE INVENTION The present invention relates to a mathematical puzzle and, more particularly, to such a puzzle made up of a number of similar three-dimensional playing pieces, each of which has a plurality of faces defining playing edges with which different indicia are associated. The invention further relates to an arrangement of such pieces into a geometrical pattern in which the indicia associated with adjacent playing edges on different playing pieces match one another in a predetermined manner.

As the role of science and technology in our society has grown, games and puzzles which test the skill of a player in mathematics or logic have become increasingly popular. For example, a puzzle of this nature marketed under the trademark Instant Insanity was quite widely accepted by the general public at the time of its introduction to the market. This puzzle comprises four separate cubes having faces of different colors. The object of the puzzle is to align all four cubes in a row with the cubes so oriented relative to one another that the cube faces defining each side of the resulting rectangular structure have a predetermined regular relationship. The arrangement on each cube of the differently colored faces is such, relative to the arrangement on the faces on the other cubes, that only one combination of specific orientations of the cubes relative to one another provides the desired solution. It will be appreciated that because each cube had six different faces and could itself be arranged in numerous orientations in space, the number of combinations of various possible cube orientations is exceedingly high. The result is that the possibility of one finding the solution via a trial and error method is quite low. However, as a practical matter, all attempts to find the solution are limited to doing so by trial and error, unless the potential solver is mathematically trained and has had experience with mathematical games so that he can discover the mathematical relationships of the cube faces to one another and use this information in arriving at the solution.

After the initial popularity of the Instant Insanity puzzle at the time it was introduced on the market, its popularity waned markedly. It is believed that one of the major reasons for this decline in market appeal is that the puzzle has only one mode of solution. That is, after a player has found the on solution, the puzzle is not, in general, any longer of interest to him. Moreover, as mentioned before, the finding of the solution is exceedingly difficult for the average player. Thus, many players have become frustrated and lost interest in the puzzle before discovering the solution.

SUMMARY OF THE INVENTION The present invention relates to a three-dimensional puzzle of the Instant Insanity type which has a plurality of different solutions, ranging from solutions which are relatively easy to find to those which are exceedingly difficult to find. Thus, the puzzle is challenging to potential players of varying skills. Because the puzzle has more than one mode of solution, the finding by the player of any one solution will not automatically take away the stimulation provided by the puzzle. Furthermore, for some of the desired configurations a logical approach is available, as distinct from the trial and error method, which when recognized can lead rapidly to the solution.

In its basic aspects, the puzzle of the invention includes a plurality of similar three-dimensional pieces, e.g., cubes, having faces defining playing edges with which indicia, such as colors, are associated. The number of difierent indicia on each face of each piece varies, depending upon the number of palying edges defined by the face and the particular arrangement or combination of the indicia represented by the edges on such face. This association of the indicia with the edges of the piece, rather than merely with the faces themselves as in the past, lends substantial versatility to the arrangement and makes a plurality of puzzle solutions, all satisfying the same puzzle criteria, possible. Most desirably, the order of the indicia associated with the edges on each face represents a different one of the mathematical combinations into which the indicia can be placed in a circular or closed order around the edges. The result is that each of the faces differs from every other face in the puzzle so that there is generally only one way in which any particular solution of the puzzle can be achieved.

The invention further includes an arrangement of such pieces into a geometrical pattern which satisfies certain criteria providing the desired plurality of puzzle solutions. Broadly, such arrangement is one in which the indicia which is associated with adjacent ones of the playing edges on different ones of the playing pieces match one another in a predetermined manner. Because the edges on each face thus play a part in defining a solution to the puzzle, solutions are made available which are not straight line solutions, i.e., solutions are possible in which the playing pieces are not necessarily aligned in a row one after another. Rather, solutions to the puzzle include various arrangements of the pieces into different geometrical patterns.

The invention will be better understood and additional features and advantages thereof will become apparent from the following more detailed description of a preferred embodiment.

BRIEF DESCRIPTION OF THE DRAWING With reference to the accompanying single sheet of drawing:

FIG. 1 is an isometric view illustrating four cubes providing a preferred embodiment of the invention, three of such cubes being shown in the proper relationship to one another to provide one solution of the invention, and the fourth cube in the process of being positioned to complete the solution;

FIG. 2 illustrates two-dimensional representations of each of the four cubes of the preferred embodiment of FIG. 1; and

FIG. 3 illustrates twelve different arrangements of the cubes of FIG. 1 which also satisfy the criteria for solutions to the puzzle of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT With reference to the accompanying drawing, a preferred embodiment of the puzzle of the invention is generally referred to by the reference numeral 11. As is illustrated, puzzle 11 is comprised of a plurality of similar three-dimensional playing pieces. There are four of such pieces, and they are in the form of cubes l2, l3, l4 and 16. Each of the cubes 12 16 has a plurality of square faces providing playing edges for defining the puzzle of the invention and the criteria by which the solutions to the puzzle are governed. More particularly, with reference to cube 12, for example, it is provided with six different square faces, one of which is referred to by the reference numeral 17. Such face 17 has four edges, 18, 19, 21 and 22, respectively. Each of said edges 18 22 is actually defined by the intersection of the face 17 with its adjacent faces making up the cube 12, with the result that each of the edges 18 22 is in fact an edge which is common to the face 17 and a respective one of the other faces of such cube.

As a particularly salient feature of the instant invention, each playing edge provided by a cube face has one of a set of indicia associated therewith. The indicia in this embodiment are different colors which are applied to the respective faces so as to be associated with the playing edges thereof. More particularly, each face on a cube is divided into differently colored quadrants, each of which quadrants is subtended by the playing edge with which the color is associated. In this particular instance, each of the colors is selected from the set of three colors yellow, red and green. For simplicity, such colors are represented in the drawing by their respective initials.

The order of the colors around the edges of each of the faces represents one of the mathematical combinations into which such colors are arrangeable in a circular or closed order. By circular or closed order is meant an order which has neither a definite beginning nor end. Since there are three colors in the set of colors, and four edges on each face, one of such colors will appear at least twice on each face.

The combination into which the colors are arranged in a circular order on each face is different for every face. That is, the order of the colors on each of the faces represents a different one of the possible combinations into which the three colors can be arranged in a circular order when taken four at a time. There are 24 different combinations in which such three colors can be so arranged in a circular order, when taken four at a time. Moreover, since each cube has six faces and there are four of such cubes in the puzzle, there are a total of 24 faces in the puzzle. The result is that the plurality of faces provided by the cubes of the preferred embodiment represents a complete complement or set of the faces. In set theory terms, this plurality of faces represents one realization of the set of all possible way of choosing, with replacement, from a set of three distinct objects, ordered sets of four objects which remain distinct under any cyclic permutation.

Most desirably, the faces are so arranged on the cubes relative to one another that the color associated with each playing edge is the same on both edges of each cube face where they intersect and have such edge in common. For example, with respect to the cube 16 shown in FIG. 1, both the upper face 23 and righthand side face 24 thereof have the color yellow in the quadrants subtended by their common playing edge 26. This feature of the puzzle reduces the possibility of confusion as to which face controls an edge, i.e., which face determines the color to be associated with the common edge.

There are various different arrangements of the cube faces relative to one another, which will result in the colors on adjacent faces associated with the common edge therebetween being the same. However, a preferred arrangement satisfying this criteria is illustrated in FIG. 2 which is a two-dimensional representation of the four cubes of FIG. 1. That is, the four figures of FIG. 2 show the six faces of the cubes they respectively represent, in the relationship such faces would have if the cubes were unfolded along edges as shown and then flattened. The same reference numerals are used in both FIGS. 1 and 2 to refer to like parts, except such reference numerals are primed in FIG. 2.

The arrangement of cube faces depicted in FIG. 2,,is an arrangement discovered by the inventor which results in the puzzle having a solution for the geometrical pattern shown in FIG. I, in which the four cubes are aligned, one after another in a row. In this face arrangement, opposite faces on two of the cubes are mirror images of corresponding faces on other cubes. One of such opposite faces on one of the two cubes is the mirror image of one of the opposite faces on the other of such two cubes. More particularly, with reference to FIG. 2, face 27 on cube representation 12 is the mirror image of face 28 of cube representation 13'. Moreover, the face on cube l3 opposite to face 28', i.e., face 29', is the mirror image of the face 31 of cube representation 14'. The face of cube l4 opposite face 31', i.e., face 32 is, in turn, the mirror image of face 33 of cube representation 16. Thus, each of the two cubes l3 and 14 have opposite faces, represented respectively at 28' and 29', and 31' and 32 in FIG. 2, which are mirror images of faces on other cubes. One of those faces which is a mirror image on cube 13, the face represented at 29, is a mirror image of one of the mirror image faces of cube 14, i.e., the face represented at 31. The other mirror image faces represented at 28 and 32' on each of the cubes l3 and 14 are, respectively, mirror images of the faces of cubes l2 and 16 represented by faces 27 and 33 in FIG. 2. This mirror image face relationship of the cubes and the criteria that the colors subtended by a common edge be the same on both faces defining such edge are the crucial concepts in determining the relative placement of each of the faces on the cubes so as to provide a solution for the FIG. I configuration.

As another important aspect of the instant invention, it includes the criteria defining the solution to the puzzle. More particularly, it includes the arrangement of the cubes into a geometrical pattern in which the colors associated with adjacent edges on the adjacent cubes are the same. Most desirably, the pattern is one in which the cubes join one another, either by edges touching or by opposed faces of adjacent cubes abutting one another. In such an arrangement, it is desirable that all adjacent playing edges of those faces on different cubes which are exposed, i.e., not hidden in the pattern, be the same color. In this connection, the criteria determining a solution to the puzzle is preferably considered met only when the criteria is satisfied by all exposed faces and edges, irrespective of the angle from which the pattern is viewed. In other words, the criteria should be satisfied even for the bottom surfaces of the pattern. However, if the faces defining an edge are all hidden within the pattern by being abutted against faces of other cubes, it is preferable that such edge need not satisfy any particular criteria. This will assure that the puzzle can have a plurality of solutions.

As mentioned previously, the solution depicted in FIG. I is possible only when the various cube faces are arranged in the relationship relative to one another described above. With reference to FIG. 1, it will be seen that all visible, adjacent edges of different cubes in the row have the same color associated therewith. The faces shown in FIG. 1 also uniquely define the orientations of all the cubes, and the adjacent edges of all exposed faces which are not depicted in FIG. 1 will also have the same colors associated therewith. That is, the edges on the rear side and the bottom of the arrangement also meet the adjacent edge criteria.

It will be further noted on considering the solution depicted in FIG. 1 that those faces of adjacent cubes which are directly opposed or abutted against one another have to be mirror images of one another before the edge color criteria is met. That is, all correspondingly opposed playing edges of those faces which are directly opposed to one another are governed by the same color. While this is true with respect to the solution shown in FIG. 1, it is not necessarily true of other solutions.

FIG. 3 illustrates twelve other solutions meeting the criteria of the invention which are possible with the preferred embodiment. Although not all faces of the cubes are depicted in the solutions depicted in FIG. 3, the faces that are depicted uniquely define the cube arrangements. The edge relationship of all adjacent exposed faces which are not depicted in FIG. 3 also meet the solution criteria and, thus, each individual figure in FIG. 3 represents a full disclosure of the solution with which it is concerned.

As mentioned previously, it is only the edges which are defined by exposed faces which should meet the solution criteria other adjacent edges need not. For example, with reference to the solution depicted at 11a, the four edges of the cubes which all meet at 36 and extend downward through the puzzle are not all defined by the same color. In this connection, it will be seen that the faces on each cube which define each of the edges meeting along the central line extending through the puzzle at 36 are hidden within the pattern. There are only three edges on each of such faces which are also defined by faces which are exposed, and therefore must meet the solution criteria.

It should be noted that for four cubes there are only thirteen different geometrical arrangements in which they can be placed joining one another with regard to the number and position of hidden faces. Thus FIGS. 1 and 3 represent solutions meeting the criteria of the invention for all thirteen of such arrangements.

The various solutions to the puzzle depicted in FIG. 1 and FIG. 3 differ in the difficulty with which they are discoverable. In most cases, the easiest solution to find is that depicted in FIG. 11 j in which none of the faces are hidden and the cubes join one another only at edges. Note that the faces which are directed inwardly toward the center of the pattern of llj also meet the solution criteria since they are exposed. The most difficult solution to find is that depicted at 110. Either a high degree of skill or a considerable time is required to find this solution which, incidently, is generally about as difficult or time consuming to find as the solution of the previously mentioned Instant Insanity puzzle. The solution of FIG. 1 as well as the other solutions depicted in FIG. 3 range in difficulty between the solutions 1 1 j and 1 1a. It will further be appreciated that the solution to FIG. 1 can be achieved easily when the mirror faces matching concept is discovered or pointed out, though its solution by the trial and error method can be extremely time consuming.

The provision of a puzzle of this nature having a plurality of different solutions assures that the puzzle will remain of interest to potential players for a considerable period of time. Moreover, because the solutions differ in the degree of skill or time required to find them, it will stimulate the interest of players of varying degrees of skill.

Although the invention has been described in connection with a preferred embodiment thereof, it will be appreciated by those skilled in the art that various modifications can be made without departing from its spirit. It is therefore intended that the coverage afforded be limited only by the terms of the claims and their equivalents.

I claim:

1. A puzzle comprising a plurality of geometrically similar three-dimensional playing pieces, each one of which has a plurality of intersecting faces providing playing edges for the piece, each of said faces having associated with each playing edge defined thereby one of a set of different indicia with the indicia associated with each respective edge of each of said faces being the same as the indicia on the other of said faces associated with said respective edge, the order of said indicia around the edges of each of said faces representing a different one of the mathematical combinations into which said indicia are arrangeable with replacement in a circular order and the total number of said different faces provided by said pieces equalling the total number of said different mathematical combinations into which said indicia are arrangeable with replacement in such a circular order; and wherein a face of a first of said pieces is the mirror image of a first face of a second of said pieces, a face of said second piece opposite said first face is the mirror image of a first face of a third of said pieces, and a face of said third piece opposite said first face of said third piece is the mirror image of a face of a fourth of said pieces, whereby said playing pieces are arrangeable in a row with the indicia associated with all adjacent playing edges of different playing pieces matching one another in a predetermined manner.

2. The puzzle of claim 1 wherein said pieces are arrangeable into a geometrical pattern in which faces of adjacent pieces are directly opposed to one another and the indicia associated with all correspondingly opposed playing edges of said faces are matched in a predetermined manner.

3. The puzzle of claim 1 wherein said pieces provide twenty-four different faces defining said edges and wherein there are three different indicia in said set of indicia, the order of said indicia around the edges of each of said faces representing a different one of the possible combinations into which said three indicia can be placed in a closed order with replacement when taken four at a time.

4. The puzzle of claim 3 wherein each of said playing pieces is a cube having six of said faces and there are four of said cubes providing said twenty-four faces.

5. The puzzle of claim 1 wherein each of said indicia is one of a set of three different colors.