ABSTRACT
Variations on the 3 × 3 × 3 cube that have been published within the last three decades are now too numerous to mention. Commercially successful puzzles nearly always spawn a host of imitations. Even if some are well conceived or even an improvement over the original, they are almost certain to languish in obscurity, since puzzle fads tend to run in cycles with no mercy on latecomer look-alikes. But we need not be concerned with that here. As an archetype the 3 × 3 × 3 cube is a superb combinatorial puzzle-simple in principle and embodiment, yet with many secret charms still lying buried inside. Perhaps we can dig a few of them out. With puzzles of this type, there is an optimum number of pieces, and as you tinker with them, you soon gain an intuitive sense of what that number is. There is no way that a four-piece version can be very diffi cult, although there is one in the next chapter that has the intriguing property of being serially interlocking, meaning that it can be assembled in one order only. The fi ve-piece and six-piece versions of the 3 × 3 × 3 cube are the most interesting. Some of the fi ve-piece designs are surprisingly confusing. The six-piece designs have the added advantage that they usually can be assembled into many other symmetrical problem shapes. (A very cleverly designed fi ve-piece puzzle might have this feature too.) In order to make a systematic study of this puzzle family, the fi rst step is to list all ways that four or fi ve cubes can be joined (as shown in Figure 63). The six-piece version of the 3 × 3 × 3 cube will be considered fi rst. For aesthetic reasons, one might prefer all the pieces to be the same size, but this is impossible, so the nearest approximation is to use three four-block pieces and three fi ve-block pieces. It is also desirable that all pieces are non-symmetrical, but this is likewise impossible, so two of the four-block pieces will have an axis of symmetry. All pieces will of course be dissimilar. Of the several thousand such combinations possible, the author tried
several that proved to have either multiple solutions or no solution, until fi nally fi nding one with a unique solution. It is shown in Figure 64. It was produced at one time as the Half Hour Puzzle . Although it was intended to construct only the 3 × 3 × 3 cube, Hans Havermann and David Barge have discovered hundreds of other symmetrical constructions possible with this set of puzzle pieces, a few of which appear in Figure 65. All of these fi gures have at least one axis or plane of symmetry, and they represent most but not all of the types of symmetry possible with this set. The cube has 13 axes and nine planes of symmetry. Two of the fi gures have one axis and two planes of symmetry. Another has one axis and one plane. All the others have one plane of symmetry only. Challenge: with this set, discover a construction with one axis and
four planes of symmetry: i.e., the same symmetry as a square pyramid. One is known. Are there more? Another six-piece version of the 3 × 3 × 3 cube is Nob’s Cube (Figure 66), by famous Japanese puzzle inventor and collector, the late Nob Yoshigahara . It likewise has only one solution. In the fi ve-piece versions of the 3 × 3 × 3 cube, there may be three fi ve-block pieces and two six-block pieces, and none need be symmetrical. The number of such possible designs must be in the thousands, and many of them are surprisingly diffi cult. One is shown in Figure 67, but readers are encouraged to experiment with original designs of their own, not necessarily using the guidelines suggested above. Throughout this book, and throughout the world of geometric puzzles in general, it is taken for granted that the sought-for solution is not only symmetrical but usually the most symmetrical possible shape-in this case, the cube. When multiple problem shapes are considered, highest priority is given to those having the most symmetry. Evidently, one of the most basic and deeply rooted instincts of mankind is a desire for
symmetry, whether in the arts, the sciences, or whatever. Trying to give reasons for so ingrained an instinct is perhaps a risky business, but here is an attempt so far as puzzles are concerned. For reasons already explained, ideally the solution of a combinatorial puzzle, by defi nition, begins with the individual pieces in the state of greatest possible disorder, meaning all dissimilar and non-symmetrical. A symmetrical solution, then, goes to the opposite extreme and does so against the natural tendency in the world toward disorder and randomness. Only the human brain is capable of doing this. Practically every human endeavor involves at least some attempt to make order out of disorder, but nowhere more graphically than in the symmetrical solution of a geometric dissection puzzle. It is the one point to which all paths lead upward and from which one can go no higher. To put it another way, the object of a well-conceived geometric recreation is usually obvious enough as to require minimal instructions. One tends to associate complicated instructions with unpleasant tasks, the defi nitive example being of course the fi ling of income taxes. Contrarily, many of life’s more enjoyable pastimes tend to require no instructions at all. Polycube pieces fi t together so naturally that some persons fi nd recreation in simply assembling random “artistic” shapes and thinking up imaginative names for them. When they don’t resemble anything that makes sense, the tendency is to call them “architectural designs.” (Does this tell us something about the present state of architectural design, or at least the public’s perception of it?)
Tetracubes Note that four cubes can be joined eight different ways. Packing a complete set of these tetracubes into a 4 × 4 × 2 box makes a neat but quite easy puzzle. There are said to be 1,390 possible solutions. They also pack into a 2 × 2 × 8 box and can be split into two 2 × 2 × 4 subassemblies.
