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Three-Square by Elap (Macey’s Pizza Parlour!)

Posted by shirleycurran on 17 December 2010

This rather strange grid was accompanied by the announcement that next year the numerical puzzles will be set for the penultimate week of February, May, August, and November, to avoid their falling on UK bank holidays. I heard a fine rejoinder. We should persuade the Cameron-Clegg powers that be to declare every weekend a bank holiday – that way, perhaps we would be rid forever of these beastly things.

We had over a foot of snow and more falling steadily so this brute was not detracting from the lawn-mowing or weeding time, but for the entire weekend, the air was, to put it decently, somewhat blue, as dead-end followed fruitless path. Mr Math worked his way through the entire thing (sorry Elap, but nothing will persuade me to use kind words for it) with a silly little plastic calculator a pencil and a mountain of paper. (OK, the choice is ours – we abandon numericals or buy decent solving equipment – but should a cross WORD – my caps. require Excel and a state of the art calculator?)

‘A primitive triangle is one in which the lengths of the sides are relatively prime’. Very kind of Elap to tell us that but what, exactly, does it mean?. I got it wrong and thought they all had to be different primes – but not so, two of the sides could share a factor, but the only common factor for the three had to be 1. Give me ‘Stripey horse (5)’ any day! I know what a scalene triangle is now and a Heronian triangle, as that seems to be another name for them, but I’d rather have the usual menu of tsetses, asti and meris.

Well, we got there, and that last sprint towards the finish line was almost (repeat, almost) fun, as entry after entry confirmed that we were right. And we sat and gazed at it. What were we supposed to do with those purple patches in the corner? We soon realized that none of the individual columns or rows provided the figures for sides of Pythagorean triplets, primitive or otherwise. We attempted to factorise the numbers to produce a pattern, we added, multiplied, subtracted, divided and cursed. We even considered cutting holes in the grid or cutting it up and reassembling it to see whether we could make three squares, an origami wren, or a Rubik cube, but to no avail this week.

READ THE PREAMBLE! I’ve been reiterating it all year long and some of last year too. That sentence ‘Solvers must insert appropriate digits in the triangular groups of shaded cells to ensure all rows and columns are thematically consistent’,  has to be giving us a hint, otherwise, any numbers we insert will be sheer guess work. The theme seems to be ‘triangles’. I feed one row into Google and seem to have hit lucky. Up comes the phone number of Macey’s Pizza Parlour in Twin Forks, Idaho. No, seriously, I see a string of vaguely familiar numbers … 1,3,6, 10, … Had it been any other row (I tried!) the answer ‘Triangular numbers’ would not have come up!

So, by an astonishing stroke of good luck, we hit on the theme but there was still hard labour to perform (with my little plastic calculator and a formula) to complete those 9 and 11-digit numbers where the first three digits, in two cases, were missing.

I have to admire Elap, all the same. How long must he have spent working through ludicrously large triangular numbers to produce the few that would give him a5 and b4, for example, – two surface areas of scalene triangles, running vertically? Despite being a complete mathematical numpty, I can recognise the remarkable feat of construction that has gone into the compilation of this grid. It must be even more complex than making nine and eleven-letter words intersect in rows and columns (as there are only ten available digits and twenty-six letters). This sort of numerical genius is out in that deep snow – way beyond my comfort area. So I should stop being churlish about being a hopeless mathematician and appreciate this numerical gem! As always, I do hope we’ll be honoured with a setter’s blog.

Warm thanks, Elap, (but please, please Mr Cameron, decree 52 bank holiday weekends next year!)


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