## Can’t You Do Division by Oyler

Posted by shirleycurran on 10 Mar 2017

The dreaded numerical! We download it and find Oyler. Well he’s the leader of the numbers brigade isn’t he? Co-editor of the new Crossnumbers Quarterly (here’s a plug – why not torture yourself an extra eight times or more every three months!) There isn’t much point my checking his right to admission to the Listener Tipplers’ Club is there since he has previously justified that with a few doubles and triples! I am fairly confident, too, that I’ll find that elusive HARE lurking somewhere in his final grid, though clearly it can’t be in a straight line (81185) as the preamble immediately tells us that each row and column contains all of the digits from 1 to 9 inclusive – a kind of sudoku without the division into 9 little mini grids and with the bonus that ‘The four entries around the perimeter are perfect squares to be deduced’.

While the other Numpty is muttering abusive words about numerical things not being crossWORDS, and beginning to fill scraps of paper with putative solutions, I run off an internet list of perfect squares with the digits 1 to 9 inclusive. I am sure there are gifted solvers who did this puzzle entirely with pencil and paper – even the list and number of factors of each solution – but I confess to using internet help and it was still far from easy. I hand over to the more numerical Numpty to explain how he proceeded.

We have 6 primes in the grid (total factors 2) and one squared prime (total factors 3). Useful, but the easiest way in seemed to involve K (form probably a**6), S (form probably a**4) and p (form probably a**6 x b**4), where the total numbers of factors implied entries 64, 16 or 81, and 5184. Happily this in turn suggested e to be 152843769. On the way! Limited options now for both B and b give the possible perimeters, and then it was a matter of inspection often for candidate numbers (such has u, xx6, of the form a x b**6). One difficulty was that a search of an apparent 6-digit prime table didn’t actually contain the needed answer, for C, 428137! A ‘number of factors’ test of the options (only 6 at the time) gave the required answer. However, despite its being a numerical, I enjoyed this one, which was attackable without hours of gazing forlornly into space, and didn’t (seem to) contain shoals of red herrings: my thanks to Oyler (from the semi-numerical Numpty)!

I hope Oyler will comment on how it could be solved without quite a lot of labour, programming or the very useful internet tools?

Later, when he sent his setter’s blog, Oyler commented “You missed the fact that m had to be 576 as it is the only 3-digit number to have 21 factors and was the first one I put in. Any number with an odd number of factors must be a perfect square”. Now I (the numberphobe Numpty) ask myself, why did I spend all those years learning every word of Shakespeare’s *Hamlet* and writing excited essays on Yeats’ and Keats’ poetry, Austen’s novels and Scott Fitzgerald’s *The Great Gatsby* when I could have been learning fascinating facts like that!

And those alphanumerical golden hares. Of course they were there – two of them!

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