Gregson found this mathematical puzzle strange and ultimately rather unsatisfying. On first reading the preamble, Gregson's heart sank, convinced it was going to be a long, slow slog with calculators and spreadsheets. Two minutes later, the coding scheme for alphabetical expressions had been guessed and the whole puzzle turned into a purely mechanical exercise with no real deduction required. It strikes Gregson that without guessing the coding scheme (and it really is only a guess, not deduction as the preamble suggests) the puzzle is probably unsolvable.

The coding scheme for words is simply to replace letters by their numerical position in the alphabet and add. To combine them in expressions, the mathematical operators are replaced as follows - addition by division, subtraction by multiplication, multiplication by addition and division by subtraction. Thus the full title encodes as

SQUARE = 81

DANCERS = 64

FOUR + STEP = 1

CODE - BY = 729

(NUMBERS/WORD) - PROCESSOR = 4096

This gives v, w, x, y and z in some as yet unknown order.

All the alphabetical clues can then be trivially converted to numbers and then next step is to deduce number bases for the rows and columns. There is only one possible base, 6, which satisfies the top row. The remaining rows and columns follow easily. Deducing the order of v, w, x, y and z is then a straightforward task to complete the numerical clues.

All in all, a puzzle which was probably quite difficult for Word Processor to construct, but was either trivially easy or impossible for solvers. On the plus side, the phrases appearing in the alphabetical clues were excellent and Gregson particularly liked LISTENER/MATHEMATICIANS.CROSSWORD and TIMES.PUZZLE/SOLVER.

Number bases for rows: 6 4 7 2 5 8 10 3 9

Number bases for columns: 5 8 2 7 9 10 4 6 3

ACROSS

1 49

3 125

5121

7 6724

9 1089

11 27

13 8

15 900

16 1225

17 9409

21 6241

23 1000

24 343

25 361

26 81

DOWN

1 216

2 169

3 1156

4 512

5 784

6 100

8 1728

10 7396

12 5329

14 2809

15 25

18 2601

19 36

20 256

22 144