Listen With Others

I decided, one day, out of pure curiosity, to see in how many different ways words could be fitted into a 4×4×4 cube, but it wasn’t necessarily for a puzzle. A 3×3×3 word cube would be too small to be interesting and a 5×5×5 one too large to have any chance of even one way of filling it. I produced a list of just over 3000 words from TEA, and graded each from 0 to 8, 0 being an ordinary, common word, and 8 being used only by Shakespeare, Spenser etc.

In the first version of the word-fitting program it soon became apparent that there were hundreds of millions of ways of filling the grids and so I restricted the words to grades 0 and 1 (i.e. no foreign, obsolete, archaic, Shakespearean words etc). The program still produced millions of filled grids and so I put in a condition that the four diagonals from the top grid through to the opposite corner of the bottom grid should also form words.

I ran the program for a couple of hours and then had another idea: what about grids which contain only ten or fewer letters? Each letter could then represent a digit. This could be the basis for a numerical puzzle with the grid ending up full of words after each digit had been replaced by its corresponding letter. A 4×4×4 cube would have 48 words, and this was about the right number of cells for a numerical puzzle. I had to disable the diagonals checks because it was very unlikely that the program would find any filled grids at all — I could always check for diagonal words afterwards (it turned out to be too restrictive).

By now it had become apparent that certain letters only occasionally appeared in any of the ‘solutions’ and so I decided to use only those words which were formed from combinations of A, C, D, E, I, L, M, N, O, P, R, S and T. The final dictionary contained 1115 words.

The program now had a sensible running time, taking just over six days to generate 370 grids. It fitted words at the rate of 50 million per second, filled grids being written to a text file. During the processing, as soon as there were more than ten different letters the program backtracked so that no time was wasted producing unacceptable grids. A second program scanned this file and analysed the grids to find what was considered to be the best one for a puzzle.

The main criterion was that at least one of the ten different letters must not appear anywhere in the first across and down words of any layer, because then a zero would have appeared there somewhere, and this would have been inelegant. The next criterion was that, just for my satisfaction, no filled grid should be similar to another one – I wanted at least ten letters different between the chosen four-layer grid and any other one. A further criterion was that no two words should have the same root, e.g. ‘alma’ and ‘alme’. The final criterion was that there must be a single-word anagram of the ten letters. Three grids were found which satisfied these criteria.

It was time to decide how one of them could form the basis for a puzzle but I forget now why I chose the one I did!

Since a word cube is involved, it seemed appropriate to use perfect cubes for the clues. The anagram of the letters was DEPILATORS – how could this be used as part a theme? In the end I decided that the letter values in the clues would be cubes with their first digit trimmed (it looked like ‘hair’ was going to be the theme) but Roger, having removed ‘trimmed’ from the preamble because the connection was not strong enough, came up with the much better term ‘shaved off’. I managed to get the letters of ‘razor’ in one of the clues and that led me to call the puzzle ‘Bearskin’. I was very tempted to call it ‘Hare’, but I thought that that in-joke had been worked to death! (Did I miss a trick here?)

The entries needed to overflow from one layer to another so that there were some decent-length numbers in the grids and not too many clues. I wanted the number of different letters in the clues to be a perfect cube, and so there had to be 27 of them. I originally had 27 clues, but that created unwanted redundancy.

The letters, when sorted, needed to include DEPILATORS and I thought it would be nice to have A WORD CUBE as well. That left eight letters with only the vowels I and U available which was quite a restriction. The only word which could have had any link to a puzzle was, fortunately, PUZZLING.

Presumably some solvers will now know that those little slits between the treads on their car tyres are called sipes!