## Listener 4738: Even Odder by Ploy

Posted by vaganslistener on 9 Dec 2022

Crumbs. It’s numerical time again. I am a real plodder when it comes to these so I tend to skip them in The Magpie, but push through in The Listener out of loyalty and because it suits my lots-of-things- lined-up-in-a-row-nicely mentality.

The first challenge is always to find The Way In. Sometimes big squares or cubes do it, or very short answers; this time it was the (rather unusual) presence of factorials. [Cue to dust down my schoolboy maths and remember that 5! = 5*4*3*2*1) They grow like Topsy and have scarcity value among numbers the size of those used in this puzzle, so looking at 11a, a factorial with five digits, there turns out only to be one, 40320 = 8!. It’s a start!

The formula in is (P1+H1)*(O1-N1)/O1^N1)! and, looking at the factors involved, (P+H) can have a value between 6 and 64. As everything before the ! has the value 8 then (O-N)/O^N must have a value less than 1, so P+H=16, (O-N)/O^N = 0.5; P+H=32, (O-N)/O^N = 0.25; or P+H=64, (O-N)/O^N = 0.125. Only the last of these produces sensible results so O=4, N=2; (O-N)/O^N = 2/16 = 0.125, and P+H = 64, and are 31 and 33 in some order.

Meanwhile 33a has to be 5! = 120 or 7! = 720. Options are P=31: (31-I-4*2)! gives (23-I)! gives I = 18 or 16 OR P=33: (33-I-4*2)! gives (25-I)! gives I = 20 or 18. At this point it looked as if it was necessary to make a leap of faith and try one of the options out. My choice was a random one but proved to be right: P=31/30, I=18/19 with R=9/8 and S=7/6 produced plausible answers for 6a, 11, 16a, 33a, 8d, 22d. There may have been a more logical path but I was happy to have struck lucky.

Then came a fairty mechanical sequence of completing equations where one letter is missing and determining its value:

G=10/11: 9a

T=26/27: 25a, 26d, 28d, 40d

B=13/12: 15a & 41a, 34d

A=15/14: 10a

L=21/20: 40a, 13d

E=23/22: 4a, 14a, 17a, 21a, 38a, 3d, 4d, 15d, 18d, 32d

U=25/24: 23a, 29a, 2d, 14d, 30d:

M=28/29: 1a, 37a, 1d, 5d, 37d

C-=17/16: 10a, 12a, 35a, 42a, 7d, 17d, 19d, 36d

I did infact partly mechanise all that by typing the equations into Excel and setting up a link to it from a table of letter and values so that as I tried out values the equations updated, and turned green if they were the right length, at which point I could see if the jigsaw was working. (I suppose that could be automated too given enough time and caffeine.)

The rather neat endgame then produced the expected sentiment in the paraphrased quotation, and happily confirmed the grid.

Thanks as ever to Ploy, and do send us more – but no more than once a quarter please!

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