Perhaps this would be a good point at which to pause for a moment and review our progress thus far. Practically every puzzle design described in this book might be regarded as a systematic dissection of some geometrical form into bits, usually identical, which are then partially recombined into puzzle pieces. The superficial perception of this strange pastime is that all of this is so that a second party can then enjoy the confusion of trying to reconstruct the original solid. In point of fact, there is often no clear dividing line as to where the design process stops and the solution begins, or who is the designer and who is the solver. They may be one and the same. In some of the plane dissection puzzles in Chapter 1, discovering the dissection is the real puzzler, after which the pattern solutions are relatively easy. In those like the Octahedral Cluster Puzzle, as presented, the design becomes the solution. In the Jupiter Puzzle, the intriguing design overshadows the straightforward solution because it is much the more interesting of the two. Some puzzles foisted upon the reader in previous chapters (if only one could write in a whisper) may not even have solutions!
It is common practice in most puzzle books to include the solutions somewhere. Perhaps some readers will be annoyed not to find them in this book. Solutions are fine when they serve some purpose. Certainly a book of riddles would be dull reading without the clever answers included, while answers to crossword puzzles may be educational. In the case of most combinatorial puzzles, including the solutions would add nothing new or interesting. There are exceptions, and note that some solutions have been included in the text. Here are four more of them:
| 1. | Karl Essley's two misplaced pieces were of course identical and triangular, but we will never know who got them, will we? |
| 2. | In the puzzling pairs of Tangram figures in which one appears to have a piece missing, the pieces are always rotated 45 degrees from one to the other. |
| 3. | The mini-Tangram set of five pieces forms seven convex figures, five of which have multiple solutions. |
| 4. | Beeler's proof of the impossibility of a 3 x 20 rectangular Cornucopia solution is to count the empty squares on either side of the piece placed and note that they are never divisible by six. |
| 5. | The reader was asked to judge which of two Cornucopia patterns was more pleasing and to identify four flaws in the other one. The following is of course not an answer but merely collective opinion. See if the reader agrees. Of the half dozen persons polled, all preferred the pattern on the right. One objectionable feature of the pattern on the left is the long horizontal line that nearly dissects the pattern. Another is the vertical line intersecting it and creating two "crossroads". (Could it be only a coincidence that a fundamental rule of good stone masonry construction is to avoid both long straight lines and crossroads for reasons of structural strength?) The two long parallel pieces at the bottom are also a distraction. A fourth flaw is that the first three flaws are all asymmetrical, creating a sense of unbalance. Having determined that this pattern is bad, it is interesting how many other objectionable features reveal themselves. The four vertical lines at the top lead the eye off the square, the T is upside down, the piece at the upper left is a pointing gun, and so on. Do you sometimes wonder what goes on (and off) inside the human mind? |
What some readers may find even more perplexing than omission of solutions is that in many cases even the designs themselves are not shown in this book but instead left for the reader to ponder. The reason of course is that the design is the puzzle, so why spoil it by giving the answer. Publishing everything known on a subject may be a good idea in some fields, such as medicine. But in recreational mathematics, a gluttony of information is probably worse than none at all. With only a few exceptions, the policy in this book has been not to include the details for any puzzle designs or solutions that have not previously been published. Instead, they have been left purposely in the dark so that the inquisitive reader may have the joy of rediscovering some of them. This book is intended to be merely a glimpse into the puzzling world of polyhedral dissections and not an open pit excavation. If every known geometrical recreation were to be dug up, extracted, and refined, would it not leave a rather barren landscape behind for future generations?
The very idea that mankind might possibly be better off with some knowledge left unpublished probably sounds as far-fetched today as did the notion a century ago that some parklands ought to be left undisturbed. The compulsion not only to publish but to be the first to do so pervades the academic world. Added to that, we now look forward with some apprehension to the day when all of this and much more will be stored and analyzed to death in some gigantic computerized retrieval system, with all the answers instantly accessible at the touch of a keyboard. But answers to what?
The use of computers is now becoming fashionable in the world of geometrical puzzles. For solving certain types of combinatorial puzzles, once the program is in place, computers can be millions of times faster than a human, and more reliable too. Several solutions mentioned in this book, such as those for the pentominoes, would probably not have been tabulated except by computer. Such exercises usually have no practical value other than simply as a programming challenge or to satisfy someone's curiosity. There is probably not a single puzzle in this book that could not be solved by computer if someone wanted to go to the trouble of writing a suitable program. Some lend themselves much more easily than others, and some would present horrendous difficulties.
Now the computer is also being used as a designer's tool. It was mentioned how the computer saves time in checking out new design ideas for the six-piece burr, and how Cutler's computer aided tabulation of burrs led to the illumination of two interesting versions that had lain dormant. The Cornucopia project was from the start an exploitation of state-of-the-art computer technology to compile a library of unique puzzle designs, which would have been impractical even just a few years earlier. A computer can even be instructed to search for most pleasing designs on the basis of certain aesthetic criteria, such as long lines and crossroads in Cornucopia solutions or difficulty index in burrs. But is this really aesthetics or pseudo-aesthetics? Is there any clear dividing line between the two, and are there any aesthetic qualities that a (non-human) computer, by definition, cannot be programmed to recognize and search for? Who knows even what is really meant by the word aesthetics?
The only significant advantage that a computer has over the human brain plus paper and pencil is blinding speed. Hence there is a tendency to program computers to solve combinatorial puzzles by brute force trial-and-error methods, whereas the human solver is always looking for clever shortcuts and usually finding them. This in itself can be a fascinating recreation. Solving geometrical puzzles by computer is rather like weeding your flower garden with a bulldozer. It may do the job quite thoroughly and rapidly, but consider for a moment all that is lost in the process, and what is the hurry in the first place?
In summary, computers are useful for solving problems that involve too much computation to be solvable by any other practical means or are just plain boring. They are misused for solving puzzles that we are either too lazy or too stupid to solve otherwise.
Shown in Fig. 173a is a portion of a checkerboard dissection with x, y coordinates added. Any single square may now be designated by its x, y coordinates, and any puzzle piece by a group of such squares. Thus, the shaded piece is 1,1; 2,1; 2,2.
Fig. 173a
Given this notation (or some other of your liking), pieces may be moved about, rotated, turned over, fitted together, etc., all with numbers alone and with no need for the physical pieces or even drawings of them. This dimensionless world of numbers is of course the only world known to electronic computers. All puzzle problems must be reduced to it before being fed in, and any geometrical figures desired must be reconstructed after digestion and disgorgement by the computer.
It is easy to add a third dimension to this scheme and thereby use it to describe polycube puzzles. The puzzle piece shown in Fig. 173b would then be described in x, y, z coordinates as 1,1,1; 2,1,1; 2,2,1.
Fig173b
Such pieces may likewise be moved about and assembled analytically. Now the question arises, given the geometrical model and its numerical representation, which is the real puzzle and which is the abstraction? To pursue that question, consider the case of higher dimensions. This numerical notation works equally well in any dimension. A three-block piece in four dimensions - w, x, y, z - might be represented by 1,1,1,1; 1,1,1,2; 1,1,2,2. Note that each square in two dimensions is adjacent to four others, represented by adding or subtracting one from any one coordinate. Likewise a cube in three dimensions is adjacent to six others, a block in four dimensions to eight others, and so on. Such higher dimension pieces may likewise be moved about and assembled into solid symmetrical solutions. The intriguing question of determining what would be considered "interlocking" or "assemblable" in four or more dimensions is left to the reader.
Now, which is the reality - numbers that we can understand (perhaps) and easily manipulate or hopelessly unimaginable hyper-geometrical models? Some Greek mathematicians, Pythagoras especially, were said to have regarded pure numbers alone as the ultimate reality in the universe and everything else as a state of mind. Modern knowledge in neurophysiology and computer science casts this profound idea in a new light. Recent developments in theoretical physics go even farther into the abstract world of numbers, where physical models actually become utterly meaningless. Perhaps more to the point, what do the terms physical and abstract really mean, if anything?
There must be very few if any other artifacts in the arts and sciences having the capability of transcending cultural barriers as do geometrical recreations. Show a dissection puzzle to persons anywhere in the world (or beyond!) and they are likely to grasp its simple message and start playing with it. Consider also their timelessness. Anyone who spends much time pondering the mysteries of the polyhedra must sense a profound kinship with past cultures that have likewise come under their spell. Are we not all Pythagoreans?
Did children of yet far more ancient times fit together clay blocks into toy pyramids or, more likely, walls and fortifications of geometrical design? Gazing in awe into the star-studded sky, one can only wonder if other cultures in other worlds ponder these same geometrical puzzles.
The educational potential of geometrical puzzles does not seem to have been very fully exploited. A fascinating course in mathematics and logic could be constructed around some of the puzzles in this book. At the same time, think of all the other related subjects that could be tied in with it - history, art and sculpture, manual arts, philosophy, psychology - perhaps even the rudiments of Freudian analysis!
Games and puzzles are closely associated. Sometimes the two words are used interchangeably, and the patents tend to be mixed together too. The most popular games have been board games, now being rivaled by video games, both of which are essentially two-dimensional. Devising a successful game that is truly three-dimensional has proven to be an elusive goal for many an inventor. There are certain practical difficulties in moving pieces about, adding or removing them in polyhedral space.
But the difficulties of polyhedral games go much deeper than that. Competitive amusements, by their very nature, tend to systematical exclude all irrelevant aspects of the game, especially aesthetics. Trying to devise a captivating game that also has much appeal to one's artistic sensibilities is almost a contradiction. Find one example if you can.
Many popular competitive games of today involve the symbolic capture, dominance, elimination, or destruction of ones opponents and their belongings. A favorite theme of video games is to accomplish this by blasting them to smithereens using the latest and most advanced space age military weaponry. What the ultimate psychological consequences of all this may prove to be, no one really knows, but it is difficult to imagine doing that artistically.
Next Christmas, why not instead give a child a hand-crafted burr puzzle or set of polyominoes. After all, someone had better begin practicing how to put all the pieces back together again.
The whole idea of adults inventing games for children needs to be questioned. I used to try to devise games for children, but I soon found that, given a box of wood scraps or other similar treasures they would quickly invent their own simple amusements which they had more fun with than any of mine.
