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Listener 4075 – Square-bashing by Arden

Posted by erwinch on 19 March 2010

Arden’s third numerical Listener since 2004.  The first, Fossil Beds, I found straightforward after a hesitant start while the second, The Latin Squire, involved finding a long Knight’s tour of the grid – this took me two hours and was a touch tedious.  The clues for Square-bashing form real words but there is no apparent theme to them while Mr and tryst each appear twice.
 
I shall only detail my route to the first crossing grid entries, which took longer than usual for these puzzles, and at first I kept on forgetting that we had to square the answers before entry:
 
11ac M / R (2) is in the range 4 to 9
20dn MR (3) is in the range 10 to 30
  
M / R = 8 / 2, 10 / 2, 12 / 2 or 14 / 2: R = 2
MR = 16, 20, 24 or 28
 
11ac TIT (2) and 22dn I + T (2) are both in the range 4 to 9: T = 1 or 3 but if T = 3 then I = 1
4dn SIR (4) is too small if I = 1: T = 1
8ac MR / S (2) is in the range 4 to 9: S = 3, 4, 5, 6 or 7
1dn S²T (4) is in the range 32 to 99: S = 6 or 7: {MR, MR / S} = {24, 4} or {28, 4}: M = 12 or 14: MR / S = 4
 
Available (green) clue answers for two-digit entries: 4 5 6 7 8 9
 
2dn TA + T (3) is in the range 10 to 31: A is in the range 9 to 20
22dn ART / S (2) = 2A / 6 (A = 15 or 18)  or 2A / 7 (A = 14)
But 2A / 7 (A = 14) = 4 so S = 6, S²T = 36, M = 12, M / R = 6 and MR = 24
8ac AS / K (2) = 90 / K (K = 10 or 18) or 108 / K (K = 6 or 12): A = 15, TA + T = 16, ART / S = 5, K = 10 and AS / K = 9 
 
4 5 6 7 8 9
 
11ac TIT (2) and 22dn I + T (2): I = 7, TIT = 7, I + T = 8 and SIR = 84
 
5dn E + LITIST (6) = E + 294L and is in the range 317 to 999: L = 3 and 15dn I + LK = 37
E + LITIST = E + 882  The third digit of the square must be 1 or 8 and the fourth digit cannot be 2, 3, 7 or 8 since perfect squares do not have these as a final digit (12ac).  There are only two fits: {E, (E + 882)²} = {6, 788544} or {13, 801025} but S = 6 so E = 13
 
This gave me my first crossing grid entries: 801025 (5dn) and 16 (8ac), which I chose to put in the right half of the grid.  After this, progress was steady and I had filled a relatively modest three sides of A4 with workings by the time just F and H remained to be assigned to 4 and 11 as determined by the 12-digit number at 1ac.  It is perhaps appropriate to remind solvers at this point that the calculator in MS Accessories can cope with 32-digit numbers and so had no trouble handling 12 digits:
 
 
1ac ADROIT + OUTPUTS – F³(OOL + S) (12)
= 32130 + 775200 – 873F³ = 751458 (F = 4) or -354633 (F = 11)
Squared = 564689125764 (F = 4) or 125764564689 (F = 11)
 
I thought this astounding.  As it happened I had the value for 1ac with F = 11 at the top of the grid and was pleased to find that this was correct after looking at the two values for 23ac: 
 
1?4041 = 184041 or 429²: 429 = 3 × 13 × 11 × 1 = LEFT
6?4656 = 614656 or 784²: 784 = 2 × 7 × 14 × 4 × 1 = RIGHT
 
This gave the final grid:
 
 
How on earth did Arden come up with this 12-digit perfect square?  Surely such things cannot have any practical use, or can they?  I used BBC Basic to see how common such numbers were up to 10 digits and then a friend showed me how to extend the range to 14 digits using MS Excel.  Any leading zeros such as 1600 / 0016 etc were ignored:
 
 
Well, not very common at all with just 0.059% of available 8-digit pairs and 0.003% at 12 digits.  I am assuming that one value for X will never pair with more than one value for Y.  Is there anyone able to extend the search to 16 digits plus?  Are examples only found where the split is an even number of digits so might we expect to see fifty pairs at 16 digits and then none again at 18?  I would doubt that there is an algorithm to generate these and suspect that they are merely one of many numerical oddities circulated among number puzzlers.  All digits in 1ac were checked  so Arden had a choice of two pairs without zeros but perhaps did not like those four sixes in a row in the alternative.  A Google search found 564689125764 on this Polish site but it appears to have no connection with Arden’s puzzle.  Of course, finding the numbers was only the beginning and you then had to write the clue using real words – I honestly would not know where to start.
 
To sum up, this was tremendous fun and dare I suggest benefiting from computer-aided setting?  Thank you Arden.
 
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