Listen With Others

There was my usual despondency as I downloaded A Defter Premier by Arden. The dreaded numerical! There was not much hope of Arden’s earning his Listener oenophile qualification with this. Then I saw the title. It had to be a premier cru! “It must be an anagram” said the other Numpty who had his pencil and paper ready to begin (yes, he solves the numerical ones that way!) TEA suggested to me that we were dealing with FREE TRADE PRIME. Well, there’s hope for us all with Brexit looming – maybe those prime wines will escape heavy duties. Numpty disillusioned me with a delighted snort. “Primes indeed, but it spells PIERRE DE FERMAT. it is going to be about his theory of prime numbers”. Cheers anyway, Arden!

Must be a hard problem, a numerical compilation! Absurd ideas, going off on a tangent, rooting around… Log in looking for topics? Prime the pump how? It’s an irrational field, sometimes, and there’s a limit to solvers’ tolerance in seeking a complex solution. How to differentiate between rational options, integrate odd ideas, converge to a real result? This was a splendid example of an approachable numerical, I think, with the corner clues being fairly easy , and strongly hinting at an ‘all prime numbers’ solution. The clues made it possible to decide on the ‘sum’ or ‘product’ variety with relative confidence, and without generating too many options to check, which can be a real ‘off putter’ for those who aren’t experts.

The other Numpty now laboriously worked his way through the clues, happily solving the ones where the clue was ‘the product’ of those two integers but muttering when the clue was the ‘sum of the integers’. Of course, knowing that only primes were being used helped enormously and, with a break for dinner, we had a full grid by mid-evening.

We suspected from the start that the two diagonals were the two symmetrically placed 9-digit numbers that share that feature (prime). I use a really valuable website (is this cheating? I really don’t know whether GCSE maths, which is the ‘supposed’ level of a solver of Listener mathematical puzzles, teaches one how to work out whether a 9-digit number is a prime  http://www.bigprimes.net/primalitytest but this does) and those two numbers were primes. I double checked and found that no other symmetrically placed pair fitted the requirement.

Clearly we needed the squares of two 5-digit numbers to produce those 9-digit primes and we both squared the available ones in the grid but as I despairingly discovered that all but two of the available solutions were way too large, the other Numpty gave his second happy shout. Of course, he had worked out that we were looking for those four diagonal 5-digit primes. What an achievement. I imagine Arden must have formed the skeleton of his grid with the two main diagonals and those four intersecting primes, then built his grid round them. I hope he will tell us how he did it!