## Listener No 4347: *Pairs* by Elap

Posted by Dave Hennings on 12 June 2015

This week, we had one of the trickier mathematical setters. Actually, they’ve all got a tricky side, so every three months I have the fear of failing to spot what is going on. Here, all we had to do was to fill a grid using letters standing for either a single square or two squares side by side. It fails to amaze me how these mathematical gurus come up with different clueing techniques.

The first thing to do was to make a list of all the possibilities for the letters:

1 | 11 | 14 | 19 | 116 | 125 | 136 | 149 | 164 | 181 | |

4 | 41 | 44 | 416 | 425 | 436 | 449 | 464 | 481 | ||

9 | 91 | 94 | 99 | 916 | 925 | 936 | 949 | 964 | 981 | |

16 | 161 | 164 | 169 | |||||||

25 | 251 | 254 | 259 | |||||||

36 | 361 | 364 | 369 | |||||||

49 | 491 | 494 | 499 | |||||||

64 | 641 | 644 | 649 | |||||||

81 | 811 | 814 | 819 | |||||||

100 | followed by the remaining squares 121–961 |

All in all, there were 76 different possibilities.

The starting point, for me at least, was 7dn *IY + YY + I (2)* which meant that Y = 1 to 9, ie 1, 4 or 9, and I < 100. From 12ac *P = IY + I + Y (2)*, I could list all the combinations of Y (1, 4, 9) with I (4, 9, 11, 14, etc) and calculate the values for P. The only values for P which were squares or square pairs were 49 (Y=4, I=9) and 19 (Y=9, I=1) and giving 7dn as 61 or 91. 23dn was therefore 13 or 10.

After that, things got a little less straightforward. However, there were fewer than normal complex possibilities to examine, so I still managed to make steady progress. It’s always satisfying when a new entry in the grid agrees with any digits already in place. I was cock-a-hoop as the grid gradually filled until in the northwest corner…!

I had 8ac *ES + B (3)* 319 (E=19, S=16, B=11) and 10ac *EEY + X (4)* 1543 (X=99). That gave 2dn *AT + U (3)* ·15. A was 44 and T 14 giving 616 and meant that U had to be 99 to give 715. However, X was already 99. Back to square 1!

Well, before that, I decided to make a list of the letters I already had in their correct sequence to see what would be spelt out. I had YIBTSELF (for 4, 9, 14, 16, 19, 25, 36). That didn’t make much sense for the first hint which described “… how to multiply each number in the larger set…”, but “BY ITSELF” was a very likely candidate. It didn’t take too long to realise that in 18ac I had overlooked the possibility of B=1.

A few minutes later my grid was full, and I was a happy bunny. A good afternoon’s work. However, all too often an ELAP grid is just the start and a tough endgame could ensue. Anyway, the full message spelt out by the letters in order was BY ITSELF GAP RDX JNUMZ. RDX could be “radix” for root, and J numbers, according to Wiki were like em>i imaginary numbers. Lawks… that looked evil!

The first thing to do was to carry out the first hint and multiply all the values in the grid by themselves: 13→169, 19→361, 35→1225, 41→1681, 43→1849. Well, the first four at least were like everything else so far — two pairs of squares run together. I continued, and managed to find twelve such numbers out of the thirty-one in the grid.

The preamble told me that I should have fifteen or sixteen of these numbers. A few more minutes passed as I teased out some of the trickier examples, such as 125² = 15625 which is 1² followed by 75², and 721² = 519841 which is 228² followed by 1².

Our second hint told us to insert a GAP. This was basically what I had been doing to identify the square pairs, and lo and behold, all the numbers thus generated matched with the others that were not of this form, such as 175 and 2281.

The odd one out was soon identified as 1771 which gave 3136441 when squared, split as 56² and 21². 5621 was not in the grid, until it was written alongside **Number** at the bottom. Luckily, imaginary numbers didn’t come into the puzzle at all.

Brilliant stuff from Elap, and in the great scheme of things, not *too* tricky after all. Many thanks for that.

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