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