## Listener No. 4412: *Analogy* by Nod

Posted by Dave Hennings on 9 September 2016

Nod’s previous three Listeners have been normal crosswords, but here was his first mathematical. Having said that, most of his Magpie puzzles have been mathematicals, the last of those, in February this year, was graded E (which I think stands for ‘Effing tough). That had me worried.

So where to start when the answers to the clues are only a fraction of what they may end up being in the grid? It looked like the grid would have to be put to one side for an hour or so. [How long!?] The letters A–Z were the first twenty-six primes and, with *x*²+*y*²=*z*², the clues were the ratios of *x* to *y* and the entry was a multiple of *z*.

The starting point was obviously one of the shorter clues, rather than one containing ten or more different letters. 9dn P:XX was it, and given that one had to be odd, P was 2 and X was 3. Well, that was a good start… except it soon became obvious that P was 3 and X was 2!

I found a list of the equations where *z* was less than 100, starting (3,4,5), (5,12,13) up to (65,72,97). There were only 16 of them. From 31ac, P(H+Y):HY, it seemed that H and Y were 5 and 11 in either order giving (48,55,73). From 4ac H must be 11, otherwise it gives 5:132. However, values of 13 and 7 also work: 31ac (60,91,109) and 4ac (13,84,85). From 10ac Y:T-Y, T = 7 or 17.

So, two options and no idea how to find which one to use. With a lot of crossword endgames, there can be a lot of staring at a grid full of words. Some people can stare at it for a few minutes before the epiphany, others several hours. Here, the grid was obligingly blank, so it was a case of clue-staring, and I found myself looking at the clues for what seemed like hours without seeing what to do next. Because of my booboo two weeks ago with *Yellow Submarine*, I actually thought that I could have another fail against my name and I wouldn’t feel too bad about it.

However, perseverance came through. From 34ac P:Q-V, Q-V = 4. From 11ac, QY < 200, so Q < 40 (since *y*≤5) and from 35ac A-V-V, V < 50. By examining all the prime differences of 4 (101-97, 83-79… 7-3), only 23-19 and 17-13 fitted. From 5dn R-A:V+V-P, V+V-P could not be prime (the *y* value in a triangle cannot be prime), so V = 19 and Q = 23.

From there on, it was plain sailing! Well, almost. It was just a question of finding clues where only one letter was unknown and fitting the known letters in. Thus, after a few hours, I had completed Stage 1, but still had a completely empty grid. Actually, I had managed to fit a few in, such as 14ac, since the *z* value was over half the maximum entry value.

Now on to Stage 2. First I had to calculate all the *x* and *y* values for every clue, especially since there were some clues that I hadn’t needed to solve to complete Stage 1. That took half an hour. Next I calculated the corresponding *z* values — another ten minutes. Finally, multiples of each *z* value had to be fitted in the grid. That wasn’t particularly difficult, but took another 45 minutes.

Thought number 1: thank goodness I finished it.

Thought number 2: how the hell did Nod come up with the idea?

Thought number 3: how did he work all the prime numbers into the clues, some of which were intricate in the extreme?

Thought number 4: that was certainly ‘Effing difficult — if not F’ing difficult!

And so, great thanks to Nod for a fantastic mathematical — just about great fun!

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