
Figure 1 shows a socalled Memory Wheel. If one looks through the window, only the three digits 000 are visible. Rotate the wheel counterclockwise over 45ยบ and the triplet 001 is in view. After a complete rotation of the wheel, all possible triplets consisting of 0's and 1's have passed the window.
It is not difficult to construct a (muchsmaller) memory wheel that produces all four duplets of 0's and 1's, or a wheel that produces all 16 quadruplets of 0's and 1's. More interesting is the fact that for making a memory wheel that produces ntuplets, the number of 0's and 1's that is needed equals the number of ntuplets to be produced. Quite difficult is the proof of the general formula that says that there are _{2}(2^{(n1)}n) different wheels to produce ntuplets. This was conjectured by K. Posthumus and proved in 1946 by N. G. de Bruijn [1]. Memory wheels were used in e.g. telecommunication to generate these ntuplets for seeker mechanisms; they were also used in constructing errorcorrecting codes.
A variation on this theme are memory wheels for ntuplets of 3 or more [different] digits.
In 1965, B. Astle published an article entitled "Pantactic Squares" [2]. Without mentioning memory wheels, he described a 2dimensional generalization of this theme. He showed the 5x5 squares as in fig. 2:
If you use a viewer as shown, twice as large as the units in the square, and place this viewer at the relevant positions on the grid, all 16 different 2x2 basic squares (figure 3) [not available] containing white and black units become visible.
In 1970, C. J. Bouwkamp, P. Janssen and A. Koene used a computer to generate all pantactic patterns [3]. They found 800 different patterns. These can be arranged into 50 classes of 16 each. Patterns in the same class can be obtained from each other by rotation, reflection and/or color inversion, as is the case with the two patterns in Figure 2.
Among these 50 pantactic patterns, only one has the property that not only the left side is equal to the right side, but at the same time the top side is also equal to the bottom side. [Graeme's note: actually, two of the 50 pantactic patterns have this property.] Hence, this particular pattern can be reproduced on a torus, producing a 4x4 (!) pantactic pattern that produces all 2x2 basic squares. This is the only real 2dimensional analogue of the memory wheel.
Bouwkamp realized that these pantactic patterns simply could be turned into a card matching puzzle. Just take the sixteen basic squares, make them form colored cardboard and there's your puzzle!
The puzzle then is to put these sixteen cards together into a 4x4 square in such a way that everywhere black and white on a side of one card match with black and white on the adjacent side of the neighboring card. Bouwkamp had one such set produced for him from aluminum squares, packed in a fine teak box. This is the puzzle as shown on page 165 of the book by Van Delft and Botermans [4].
Bouwkamp's puzzle pieces in their teak box, as shown in the picture, have a very special property: Each piece, when turned upside down, shows in red again one of the basic squares. Bouwkamp combined the red basic squares in such a way with the green squares that, whenever the green sides show a pantactic solution, the red sides show a solution of the corresponding dismatching puzzle. What's that, you may ask. In a matching puzzle, white must match with white on the neighboring card and black (or rather green in this case) must match with green. In a dismatching puzzle, different colors must match, so white "matches" with red (or black) and red "matches" with white. Also, when the green pieces are put in the box so as to give a solution for the dismatching puzzle, turning all pieces upside down produces in red a solution for the matching puzzle.
It was Bouwkamp's merit to turn the pantactic squares into this fine pantactic puzzle. He too was the first to realize that there are two different transformations to turn any matching solution into a dismatching solution. Of course, his dismatching puzzle still has the same 800 solutions as there are pantactic patterns. Again, these 800 solutions can be classified into 50 classes of 16, where all sixteen solutions in one class show the same pattern, at least after rotating, reflecting and/or color inversion of such a pattern. The onetoone relationship between matching and dismatching solutions, makes clear that the same 50 classes of 16 solutions appear for the dismatching puzzle.
Recently, I saw two puzzles, put on the puzzle market a few years after the publication of Van Delft's book [4]. One by Peter Pan Playthings, the other by Regev Magnetics. Neither of them gives an indication that among the 16 pieces (the basic blocks), equallooking pieces must be placed in different rotations and both make the incorrect claim that the puzzle has 50 solutions.
It is too clear that these two producers did not understand the idea behind Bouwkamp's puzzle. They just copied what they saw in Van Delft's book. Of course, both puzzles have only onesided pieces: Van Delft did not mention the backside in his book: Bouwkamp did not tell him why the pieces were twosided! Only the green side was photographed for the simple reason that green looks better on the photograph! Who would believe that the puzzles as published by Peter Pan and Regeve [sic] were not copied from what was shown in the book? Neither producer asked Bouwkamp for his permission to produce the puzzle.
On February 1, 1975, the Dutch newspaper Trouw published puzzle #572 in the column "Niet Piekeren maar Puzzelen". This puzzle was invented by J. J. M. Verbakel [I think the name Verbakel should be spelled with a "schwa" (upsidedown "e") in place of the a]. Named "Digitale Tegels" (Digital Tiles), this puzzle too consisted of 16 square pieces that are not allowed to rotate (figure 4).
Again, aim of the puzzle was to put the 16 pieces together into a 4x4 square, so that matching sides show the same color. Instead of 800 solutions, this version has no less than 2,765,440 solutions. These too can be classified into 172,840 classes of 16 closely related cases. Again, it is easily seen that putting the pieces together so that the sides of the pieces dismatch, has the same number of solutions.
Because of the huge number of solutions, the puzzle as published in Trouw was not just to match the 16 pieces, but added more constraints in 6 different ways, most of them by putting 4 named pieces in given positions. More interesting was the constraint to assemble the pieces in such a way that the numbers on the cards produce a magic square.
It is left as a challenge to the reader to find one special solution to Verbakel's puzzle. In that solution, along each horizontal and also along each vertical line, all 8 matching segments have the same color. Not only "inside" the 4x4 square, but in the same way the left side must match the right side and the top must match the bottom side with 8 equally colored segments.
For the readers interested in etymology: the word "pantactic" was first used by B. Astle [1]. It is derived from two Greek words: "panta", which means "all", and a conjugation of the verb "tattoo", from which the English word "tactics" was derived. In this context "tactic" is to be understood as a "formation". So "pantactic" means that all possible formations are present.
Jacques Haubrich, Eindhoven
References
[1] Bruijn, N. G. de, "A Combinatorial Problem", Indagationes Mathematicae, Akademie van Wetenschappen (Amsterdam), 8 (1946), pp. 461467
[2] Astle, B., "Pantactic Squares", Math. Gazette, 49 (1965), 144152
[3] Bouwkamp, C. J., P. Janssen, A. Koene, "Note on Pantactic Squares", Math. Gazette, 54 (1970), pp. 348351
[4] Delft, Piet van, Jack Botermans, Puzzels uit de hele wereld, Spectrum Hobby, 1978 (translated in many languages)
http://www.mathematik.unibielefeld.de/~sillke/PUZZLES/pantacticsquares/
Scanned images of a Jacques Haubrich article about Pantactic Squares.
The webmaster and author of this Math Help site is Graeme McRae.