GB2373738A

GB2373738A - A cube puzzle

Info

Publication number: GB2373738A
Application number: GB0107509A
Authority: GB
Inventors: Lucas Mellinger
Original assignee: Individual
Current assignee: Individual
Priority date: 2001-03-26
Filing date: 2001-03-26
Publication date: 2002-10-02
Also published as: GB0107509D0

Abstract

A puzzle comprises a cube made up from eight cuboids in which each of the three exposed faces of each cuboid combine with the exposed co-planar faces on the adjacent cuboids to present unified images on each of the six faces of the cube, or whose unified images combine to represent a single three dimensional image. Preferably the cube is of dimension (a+b), (c+d) and (e+f) where a+b = c+d = e+f. Optionally (a+b), (c+d) and (e+f) are selected from the set of (0.5,0.5), (2/3,1/3) and (( ! 5-1)/2,(3- ! 5)/2). The images may be single colours, numbers, one to 6 as arranged on a die, a pattern, stripes, pictures or words.

Description

Puzzle The present invention relates to three-dimensional puzzles and in particular to cubic three-dimensional puzzles.

There are a number of puzzles known which test manual dexterity and mental agility, and these are of varying degrees of complexity. Some of these puzzles concern the ability of the user to organise fragmented images of two dimensions into a single unified image of two dimensions, for example jigsaws. Other puzzles concern the ability of the user to arrange a variety of geometrical shapes into a single geometric shape. However, there is no puzzle which combines these two skills and requires the user to organise fragmented parts of a three dimensional shape into a unified geometric shape, which, when assembled, bears unified two-dimensional images on all the exposed surfaces.

It is therefore an object of the present invention to provide a three-dimensional geometric puzzle which also requires the user to organise fragmented images on the surfaces of the puzzle to produce a solution in which unified two-dimensional images appear on all the exposed surfaces.

It is a further object of the present invention to provide a three-dimensional puzzle which has two separate sets of two-dimensional image solutions.

It is another object of the present invention to provide a three dimensional puzzle which also operates as an educational aid.

According to the present invention, there is provided a puzzle comprising a cube made up of eight cuboids in which each of the three exposed faces of each cuboid combine with the exposed co-planar faces on the adjacent cuboids to present unified images on each of the six exposed faces of the cube, or whose unified two-dimensional images on the exposed surfaces combine to represent a single three-dimensional image.

Preferably, the puzzle has a second set of images which are presented when each of the eight cuboids is rotated to expose the three previously hidden faces, the cuboids not necessarily being in the same relative positions. The cuboids may, in addition to being rotated to expose the other three faces, be moved relative to each other. The puzzle therefore contains two separate solutions.

In a more complicated embodiment of the invention, the second set of images may be the same as the first set, although not necessarily in the same relative positions i. e. a red side may be opposite a yellow side in one solution, but may be opposite a blue side in the other solution. The solutions to each puzzle would therefore be more difficult to obtain as it will not be known whether the section of image A is from the first solution to the puzzle or from the second and therefore whether the face including this section should be exposed or hidden.

In a simpler version of the puzzle, the second set of images is completely different from the first set and it is therefore easier to determine the orientation of the pieces to assemble each solution.

The cubic puzzle of eight pieces will generally be formed by making three cuts in the cube in mutually perpendicular directions along the axes of the cube. Each side will therefore be split into two parts-a and b in a first direction, c and d in a second perpendicular direction and e and f in the third mutually perpendicular direction.

Naturally a+b=c+d=e+f=l to form a cube of unit dimension. The pairs of dimensions (a, b), (c, d) and (e, f) are preferably selected from the set of values (0.5, 0.5), (2/3,1/3) and ( (5-1)/2, (3-5)/2).

Preferably a=c=e and b=d=f. In this case the three cuts will be made in similar positions along each of the three mutually perpendicular axes and there will be more than one of some dimension of piece. Since these identically sized pieces can go in more than one position and still make up a cube, this increases the number of variables and makes the

puzzle harder. If awcxe and bwdtf then there will be eight different cuboids and there will be substantially fewer geometric solutions to the puzzle of forming the cube.

Optionally, a=c=e=1/3 and b=d=f=2/3. Alternatively, a=c=e= (5-l)/2 and b=d=f= (35)/2 in what is known as the golden section where the relationship between the smaller length to the larger length is the same as that between the larger length and the whole.

In another alternative embodiment, a=b=c=d=e=f=0. 5. This cube therefore comprises eight identically sized smaller cubes which maximises the number of possible geometrical arrangements of the eight pieces to assemble the overall cube. Since all eight pieces are of the same size, they can each occupy any of the eight positions in the overall cube in any orientation. Generally, each cuboid has eight different orientations to show different combinations of three external faces (see figure 11). The total number of different ways in which the eight cubes of this embodiment can be arranged is therefore 88 = 16, 777,216.

The two-dimensional images on the external faces of the cube when the puzzle has been solved may take any desired form. Optionally, the images may be faces of single colours.

Alternatively, the images may take the form of the numbers one to six arranged as on a conventional die or in any other arrangement. Another alternative is some form of pattern such as a striped pattern or another geometrical pattern such as spirals. Still another alternative is a series of pictures or words or a combination of the two. Another alternative is a series of unified two-dimensional images on the exposed faces of the cube combining to represent a single three-dimensional entity. Examples include a flat roofed building and a postal package secured with string.

Preferably the second set of images are of a similar nature to the first, for example if the first set comprises faces of single colours the second set are all faces of a single colour as well. In the case of the numbers of a die, the second set of images is optionally a different set of numbers, for example the numbers seven to twelve. As indicated above, the puzzle is considerably harder if the second set of images is the same as the first.

The present invention may be put into practice in various ways and a number of specific embodiments will be described by way of example to illustrate the invention with reference to the accompanying drawings, in which: Figure I shows a cube of side lengths (a+b), (c+d) and (e+f) where awcwe and bodxf ;

Figure 2 shows a cube where a=1/3 and b=2/3 ; Figure 3 shows a cube where a= (5-l)/2 and b= (3-5)/2 ; Figure 4 shows a cube where a=b=0. 5; Figure 5a shows a perspective view from above of a cube where the images are faces of single colours; Figure 5b shows a perspective view from below of the remaining three faces of the cube of figure 5a; Figure 6a shows a perspective view from above of a cube where the images are the numbers one to six arranged as on a conventional die; Figure 6b shows a perspective view from below of the remaining three faces of the cube of figure 6a; Figure 7a shows a perspective view from above of a cube where the images are a striped pattern; Figure 7b shows a perspective view from below of the remaining three faces of the cube of figure 7a ; Figure 8a shows a series of six picture images with associated words based on the vowel "a"which could be used on a cube of the present invention; Figure 8b shows a series of six picture images with associated words based on the vowel "o"which could be used on a cube of the present invention; Figure 9a shows a perspective view from above of a cube where the two-dimensional images on each face combine to form a three-dimensional image of a package secured with string; Figure 9b shows a perspective view from below of the remaining three faces of the cube of figure 9a;

Figure 1 Oa shows a further embodiment of the cube of figure 3 in which the top layer of cuboids has been rotated through 900 clockwise relative to the bottom layer ; Figure 10b shows a still further embodiment of the cube of figure 3 in which the top layer of cuboids has been rotated through 1800 relative to the bottom layer ; Figures 1 la to 11h show a cube and the eight different sets of the three exposed faces a cuboid can exhibit; and Figures 12a and 12b show leaflets giving the two solutions to an embodiment of the present invention.

Figure 1 shows a cube according to the present invention broken up into eight cuboids. The cube is broken up by means of cuts in the three mutually perpendicular directions of the axes of the cube. In this general case, the position of each cut along the length of each

side is different so that ace and bdf. This produces eight cuboids of different volumes ace, ade, acf, adf, bce, bde, bcf and bdf. This puzzle would not be too difficult to solve since the irregularity of the pieces would mean that geometrically there are only a couple of solutions to the puzzle. This would effectively reduce the puzzle to the second element of the present invention, namely the organisation of fragmented images on each of the exposed faces into unified images of two dimensions.

Generally, therefore the puzzle will be of the form where the cut is in the same relative position on each side such that a=c=e and b=d=f. In this case you will have a number of pieces of the same size and there will therefore be some choice in the positioning of each element. This choice will be affected by the two dimensional images on each of the exposed faces. Examples of such puzzles are shown in figures 2,3 and 4.

An advantage of such puzzles is that the volume of the overall cube is (a+b) 3 and of the eight cuboids produced by the cuts one each will have a volume of a3 and b3 respectively and three each will have volumes of ab and ab2. This puzzle may therefore be used as an educational aid to help teach the equation (a+b) 3 = a3 + 3a2b + 3ab2 + b3. Similarly, looking at the top layer of cuboids in, for example, figure 3 it is noted that the area of the top surface is (a+b) 2 and that the areas of the respective faces of the four cuboids are one each

of a2 and b2 and two of ab. Thus the puzzle may be a learning aid for the equation (a+b) 2 = a2 +2ab + b2.

The puzzle may also be used as a geometric proof of Fermat's Last Theorem"If n is a whole number greater than two, there are no whole numbers a, b, c such that an + bn = cn".

The puzzle shows that this is true for n=3, but the solution applies equally for values of n greater than three as shown below. a'+b'= c' < = > a'=c'-b' (1) Let b be a whole number smaller than c and smaller than a c = b + k (2) where k is an integer.

Substituting (2) in (1) = > a3 = (b + k) 3-b3 (3) As set out above, the expansion of (b + k)'= k 3+ 3bk2+ 3kb'+ b' (4) Substituting (4) in (3) = > a3 = k3 + 3bk2 + 3kb2 + b3-b3 = k3 + 3bk2 + 3kb2 (5) Looking at the puzzle, removing the smaller cube it is clear that, regardless of the size of b, a3 can never be a complete cube of a whole number a.

For example, if a = 7 there are three different canonical expansions of the eight cuboids : 73 = (6 + 1) 3 = 6'+ 3 x 62x 1 + 3 x 6 x 12 + 13 73 = (5+2) = 5 + 3 x 52x 2 + 3 x 5 x 2'+ 23

7'= (4+3)'= 4'+3x4'x3+3x4x3 2+33 If the final cuboid (13, 23, 33) is removed, the final decomposition of the above cube 73 is deficient. Therefore a3 k + 3bk2 + 3b2k for a whole number a.

For n > 3, the same proof inherently applies. Generalising equation (5) an = (k + b)"-b"= kn + c, kn-'b + c,n-2b2 +... + c2k2bn-2 + Clkbn'l + bn-bn (6) Once again, the removal of bn from the canonical decomposition of an, means that, regardless of the value ofb, this equation can't be true for a being a whole number.

Figure 2 shows a cube where the cut on each side is two thirds of the way along the side. This produces cuboids where one dimension is twice the value of the other. In this case there is a special relationship between the volumes of the individual cuboids. If a=1/3 and b=2/3 as shown, the a3 cube will be exactly half the volume of each of the three a2b cuboids, each of which is in turn exactly half the volume of each of the three ab cuboids, each of which is exactly half the volume of the b3 cube.

This relationship makes the puzzle harder since the positions of key features in different sized cuboids will be similar in relation to the edge. Referring to figures 6a and 6b which are shown on a cube divided according to the golden section, the dot which represents the number one can be distinguished from the two separate dots which form the number two by looking at the distance of the dot from the edges of the piece. The distances will differ.

This same exercise would not be distinguishing if the division along the sides were 1: 2 since the dot would be the same distance from the two edges in both cases.

Figure 3 shows a cube where the cut has been made such that the cube is broken up according to the so-called golden section. This is defined as a line cut in such a way that the smaller part is to the greater part as the greater part is to the whole. In practice this

means that the larger dimension a= (5-1)/2 and the smaller dimension b= (3-5)/2. As required, b/a= (5-l)/2. Figure 4 shows a cube where each cut is made at the midpoint of each side such that a=b=0. 5 and the cube is broken up into eight identical cubes of a length half that of the overall cube. As indicated above, the puzzle with eight identically sized cubes provides the greatest number of possible geometrical solutions since all eight pieces are equally interchangeable in their relative positions. As well as being moveable relative to each other, each cube can have eight different orientations to expose three different faces.

Figures 11 b to 1 li show the eight different sets of the three exposed faces a cuboid can exhibit. Numbering the faces of the cube one to six as shown in figure Ha and having face one positioned at the top, there are four different combinations for the remaining two exposed faces (see figures 1 b to 1 e). With the face numbered two at the top there are only two different combinations (figures 1 If and llg) as the other two combinations which include the face numbered one have already been shown in figures 11 b and He.

With the face numbered three at the top there is only one new combination as shown in figure 1 Ih, the remaining three combinations having been shown in figures 11 b, 11 c and 1 If respectively. Finally with face numbered four at the top there is one new combination as shown in figure 1 lui. The other three combinations using the face numbered four are shown in figures He, lid and 1 oh. All four combinations for the sets including the faces numbered five and six have already been disclosed.

The embodiments shown in figures 2 to 4 are simply examples of the sort of divisions along each axis which may be made and are not intended to be limiting. Any ratio of cut may be used to suit the individual situation, for example a=3/5, b=2/5 ; a=3/10, b=7/10; a=l/4, b=3/4; etc.

The images on the faces of the cube may take any desired form and may be varied to suit the level or interests of the user. Examples of possible images include, but are not limited to the following examples as shown in figures 5 to 9. These examples have been shown

with the cube broken up according to the golden section but, as indicated above, any other split may be used.

Figures 5a and 5b show a cube where each face of the cube comprises a single different colour. For example, one set of six faces 1, 2,3, 4,5, 6 may be coloured with the primary and secondary colours of red, blue, yellow, violet, green and orange respectively. A second set of colours for the second solution to the puzzle may be different, tertiary hues of red, blue and yellow. In the simplest arrangement of the cuboids of whichever size, the solution showing the second set of colours may be obtained by simply rotating each cuboid through 900 about each of the three perpendicular axes. In a more difficult arrangement, the solution showing the second set of colours will involve the movement of the cuboids relative to each other in addition to the rotation.

Another set of images which may be displayed on the faces of the assembled cube are the numbers one to six arranged as on the faces of a traditional die (see figures 6a and 6b).

The second solution to this numerical version of the puzzle may be the numbers seven to twelve arranged such that the opposite sides add up to nineteen (seven opposite twelve, eight opposite eleven and nine opposite ten).

In a more complicated embodiment of the puzzle, the second solution may be the numbers one to six again arranged as on the faces of a traditional die. In this case, it will be difficult to know whether each piece is arranged the correct way round until the puzzle is completed as you will not be certain whether the spot for the number one, for example, is from the first solution of the second. Of course, if you wanted to make this puzzle slightly easier, you could colour the spots for the two solutions differently, for example red for the first solution and blue for the second.

Figures 7a and 7b show a cube where the image is a striped pattern. In the example shown in figures 7a and 7b, the stripes are of uneven width and are of alternate colours, for example red and white. One colour of stripe is indicated by the numeral 10 and the other colour is indicated by the numeral 11. It can be seen that in this example, the

diametrically opposite comers 12 and 13 each have a triangle of the same colour on all three exposed faces and this may be an aid to solving the puzzle. Of course, the stripes could be of equal width and the number of them could be altered. More than two colours could also be used to form a particular pattern.

As indicated above, the image could be pictures or words or a combination of the two. In one preferred embodiment a combination of pictures and words can be placed on each side, enabling the puzzle to be used as an educational tool. Examples of combinations of pictures and words which could be used are shown in figures 8a and 8b. In these particular examples, one set of images concentrates on the vowel"a" (see figure 8a) and the other set concentrates on the vowel"o" (see figure 8b). These two sets could be the two solutions to the puzzle. Of course, the pictures do not have to be of animals and could be of any subject. The puzzle could also be sold with some or all of the faces left blank for the customer to apply images or letters of their own choice to customise the puzzle to the individual.

Figures 9a and 9b show a cube where the two-dimensional images on the faces of the cube combine to form a single three-dimensional image of a postal package secured with string. Of course other three-dimensional images could be shown on the cube such as, for example, a flat roofed building, a cereal packet, a book etc.

In the simplest arrangement of the cuboids to form the desired images, the image will be imposed on the cube before it is cut and the cube is then cut in the three different planes.

This means that the completed puzzle will have the three cuts all lined up once again as shown in, for example, figure 3. In a more complicated arrangement, after the initial cube is cut in the three different planes, the top layer is rotated clockwise through 900 relative to the bottom layer and then the images are imposed on the cube. This means that there will only be two of the cuts lined up along the lines 20 and 21 in figure 10a when the puzzle is completed. In a further extension to the complexity, the top layer may be rotated by 1800 relative to the bottom layer and then have the images imposed on the cube. This

results in there being only one of the initial cuts lined up along the line 23 in figure 10b when the puzzle is completed. As is clear from the above, there are a number of different embodiments of the present invention which can be targeted at different levels and interests of user. The puzzle may be used as an educational tool at various levels of learning and the images used and the arrangements of the cuboids to form the cube can be adjusted to suit the appropriate user.

Of course, the puzzle can be used solely as a toy and it is not limited to educational uses.

The puzzle, when marketed, is preferably accompanied by the solution to the puzzle in the form of a cross-shaped leaflet. On one side of the leaflet are shown the six faces of the first solution of the puzzle, with the faces being juxtaposed in the correct relative positions such that when the leaflet is folded along the vertices, the assembled cube is shown. On the other side of the leaflet, the second solution to the puzzle is shown. The leaflet preferably also includes indications of where the cube is broken down into the cuboids. This extra feature may be particularly advantageous in the puzzle where the images are the numbers one to six arranged as on the faces of a traditional die.

Figures 12a and 12b show one such leaflet for a puzzle in which one solution is the numbers one to six as on the faces of a conventional die, and the other solution is a set of faces each with seven spots on them. The spots indicated by the reference 20 in Figure 12b are of a different colour to the remaining spots of both the first and second solutions, for example red compared to yellow. Each spot 20 counts double compared to the remaining spots. The second solution is therefore a cube representing the numbers seven to twelve with sides seven and twelve, eight and eleven and nine and ten respectively opposite each other such that opposite faces add up to nineteen. The leaflet can be folded along the vertices and show, at a glance, the correct relationship between the eight cuboids to generate either of the two solutions.

Claims

Claims 1. A puzzle comprising a cube made up of eight cuboids in which each of the three exposed faces of each cuboid combine with the exposed co-planar faces on the adjacent cuboids to present unified images on each of the six exposed faces of the cube, or whose unified two-dimensional images on the exposed surfaces combine to represent a single three-dimensional image.
2. A puzzle as claimed in claim 1, in which a second set of images are presented when each of the eight cuboids is rotated to expose the three previously hidden faces, the cuboids not necessarily being in the same relative positions.
3. A puzzle as claimed in claim 1 or claim 2, in which the cube is of dimension (a+b), (c+d) and (e+f) where a+b=c+d=e+f.
4. A puzzle as claimed in claim 3, in which (a, b), (c, d) and (e, f) are the same or different.
5. A puzzle as claimed in claim 4, in which (a, b), (c, d) and (e, f) are selected from the set of (0.5, 0.5), (2/3,1/3) and ( (5-1)/2, (3-/5)/2).
6. A puzzle as claimed in claim3, in which a=c=e and b=d=f.
7. A puzzle as claimed in claim 6, in which a=b=0. 5.
8. A puzzle as claimed in claim 6, in which a=1/3 and b=2/3.
9. A puzzle as claimed in claim 6, in which a= (5-1)/2 and b= (3-5)/2.
10. A puzzle as claimed in any preceding claim, in which the images are faces of single colours.
11. A puzzle as claimed in any one of claims 1 to 9, in which the images are numbers.
12. A puzzle as claimed in claim 11, in which the numbers are one to six arranged as on the faces of a conventional die.
13. A puzzle as claimed in any one of claims 1 to 9, in which the images are a pattern.
14. A puzzle as claimed in claim 13, in which the pattern is a series of stripes.
15. A puzzle as claimed in any one of claims 1 to 9, in which the images are pictures.
16. A puzzle as claimed in any one of claims 1 to 9, in which the images are words.
17. A puzzle as claimed in any one of claims 1 to 9, in which the images are pictures which combine over the exposed faces to represent a single three-dimensional image.
18. A puzzle as claimed in any one of claims 2 to 17, in which the second set of images are the same as the first set of images.
19. A puzzle as claimed in any one of claims 2 to 17, in which the second set of images are different from the first set of images.
20. A puzzle as claimed in any preceding claim, in which the puzzle may be used as a geometric proof of Fermat's Last Theorem
21. A puzzle constructed and arranged substantially as herein and specifically described with respect to and shown in Figs 1 to 7b or 9 to 10 of the accompanying drawings.

20. A puzzle as claimed in claim 19, in which the second set of images is of a similar nature to the first set of images.

21. A puzzle constructed and arranged substantially as herein and specifically described with respect to and shown in Figs 1 to 7b or 9 to 10 of the accompanying drawings.

Amendments to the claims have been filed as follows 1. A puzzle comprising a cube made up of eight cuboids, at least one of which is not a cube, in which each of the three exposed faces of each cuboid combine with the exposed co-planar faces on the adjacent cuboids to present unified images on each of the six exposed faces of the cube, or whose unified two dimensional images on the exposed surfaces combine to represent a single three dimensional image.

2. A puzzle as claimed in claim 1, in which a second set of images are presented when each of the eight cuboids is rotated to expose the three previously hidden faces, the cuboids not necessarily being in the same relative positions.

3. A puzzle as claimed in claim 1 or claim 2, in which the cube is of dimension (a+b), (c+d) and (e+f) where a+b=c+d=e+f.

4. A puzzle as claimed in claim 3, in which (a, b), (c, d) and (e, f) are the same or different.

5. A puzzle as claimed in claim 4, in which (a, b), (c, d) and (e, f) are selected from the set of (0.5, 0.5), (2/3,1/3) and ( (5-1)/2, (3-5)/2).

6. A puzzle as claimed in claim3, in which a=c=e and b=d=f.

7. A puzzle as claimed in claim 6, in which a=1/3 and b=2/3.

8. A puzzle as claimed in claim 6, in which a= (5-l)/2 and b= (3-5)/2.

9. A puzzle as claimed in any preceding claim, in which the images are faces of single colours.

10. A puzzle as claimed in any one of claims 1 to 8, in which the images are numbers.

11. A puzzle as claimed in claim 10, in which the numbers are one to six arranged as on the faces of a conventional die.

12. A puzzle as claimed in any one of claims 1 to 8, in which the images are a pattern.

13. A puzzle as claimed in claim 12, in which the pattern is a series of stripes.

14. A puzzle as claimed in any one of claims 1 to 8, in which the images are pictures.

15. A puzzle as claimed in any one of claims 1 to 8, in which the images are words.

16. A puzzle as claimed in any one of claims 1 to 8, in which the images are pictures which combine over the exposed faces to represent a single three-dimensional image.

17. A puzzle as claimed in any one of claims 2 to 16, in which the second set of images are the same as the first set of images.

18. A puzzle as claimed in any one of claims 2 to 16, in which the second set of images are different from the first set of images.

19. A puzzle as claimed in claim 18, in which the second set of images is of a similar nature to the first set of images.