mmar@sphinx.UChicago.UUCP (Mitchell Marks) (08/20/85)
After looking over the examples, I realized I should say more about what
a word cube is.
As with 2-dimensional word squares, the words along any one dimension
all run in the same orientation. Even in the symmetrical squares & cubes,
the correctness is not preserved under 180 degree rotation, or reflection
through an orthogonal line or plane.
All rows or columns must be good words, in the fixed orientation.
There is no margin for garbage, no blanks, no wildcards. Shorter contained
words are of no significance; all that matters are words of the full length
(the order of the cube), filling some row or column.
For the non-symmetrical cuubes, all words must be unique. This gives
3 * (N ^ 2) words for an order-N square: 108 for order-6, 147 for order-7.
Thus, each layer must be a good square on its own, and each column through
the layers must be a word. Put differently, each layer must be a good square,
no matter which dimension you slice the layers along. (Non-symmetrical squares are harder.)
The syymmetry expected of symmetrical cubes can be defined in a number
of ways. In the case of 2-dimensional squares, reflection across the
upperleft-lowerright main diagonal must preserve the square exactly.
(Note that the SATOR-AREPO-etc square has more symmetry than required in
the general case.) In three dimensions, each layer should be a symmetrical
square in the sense just given; and this should remain true even if the
cube were reassembled and sliced in either of the two other ways. Another
way of putting it is that the letter found at (x,y,z) must also appear
at all permutations of those coordinates.
Thus, if these were real words, the following would be a
symmetrical order-3 cube:
1 A B C 2 B D E 3 C E F
B D E D G H E H I
C E F E H I F I J
It uses the `words' ABC BDE CEF DGH EHI and FIJ. This is the maximal
variation allowed within the symmetry (I didn't use the same letter
anywhere that a new one was allowed). Notice that none of these are
palindromes; finding palindromes sounds like a promising element in
making a symmetrical square or cube, but it isn't (unless you're
trying for the extra symmetry of the SATOR square).
following the rules of symmetry is a pain, but the problem is
still easier than for the non-symmetircal case. Where that order-3
example used six words (and they could have been re-used more), a
non-symmetrical order-3 cube would require fitting together 18 words.
--
-- Mitch Marks @ UChicago
...ihnp4!gargoyle!sphinx!mmar