Listen With Others

This is no blog.

A genuine reminiscence of the setting process would almost certainly read “It took a long time to produce the grid.” Hardly the riveting read of the beautifully anecdotal accounts that normally litter these pages. Instead the following could best be described as an invention cobbled together through the haze of history. Looking back I think this might be how it was done – probably.

The chief difficulty in devising A Defter Premier was undoubtedly filling the grid in an acceptable way. It would not have been fair to solvers to ask them to check the primality of large numbers unless many of those that needed checking were obviously composite or were of the form 4N-1, which cannot be split. Multiples of 2, 3 and 5 can easily be spotted so the idea was to include as many of these as possible to eliminate rows and columns.

As solvers would be searching the grid for candidates it was necessary to consider rows and columns both forwards and backwards and attempt to guarantee at least one of each symmetrical pair could be ruled out with relative ease. This would then force the solver to look at the diagonals and probably trust that these are truly primes of the form 4N+1.

It was a deliberate decision to have the 9-digit primes that required splitting ending with a 7, as this forces their square parts to end with 1 and 6. Keeping the 4s and 6s in the grid to a minimum would therefore also assist solvers in the final hunt.

As it was unnecessary to the solving process I wrote a computer program to examine whether a particular number could be split into two perfect squares which allowed checking of every row and column. After a great deal of trial and error this finally produced a grid where virtually every row and column, forwards and backwards, is obviously composite and none can be split into two constituent squares.

Having arrived at a suitable grid cluing was relatively easy. By Fermat’s 4N+1 Theorem each entry has a unique division into squares. Clues, therefore, would combine the appropriate square roots to celebrate that unlikely fact.

Entirely using products for the clues would have been both tedious and simple, whereas using sums would give too many ambiguities, so a mixture was chosen. Some products that needed working out were included to get the solver started and then sums for a bit of variety. Having sussed out what is going on the solver can use the size of the clue and whether it is odd or even to decide if it is a sum or product.

The ultimate aim of the puzzle was a hope that some solvers would discover this extraordinary theorem.