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Posts Tagged ‘Word Squares’

‘Word Squares’ by Elap

Posted by Encota on 8 Dec 2017

A very nice puzzle – thank you Elap!  The initial Preamble was pretty daunting and, combined with the terseness of the clues has perhaps set a new ‘High Score’ for:

(number of characters in Preamble)/(number of characters in Clues)

The numerical deductions took a while but it was all worth it.  I was briefly thrown when my electronic Chambers didn’t give SAIRS as a plural of SAIR but the BRB definition fully backed up its use – phew!

2017-11-20 10.22.42

And Elap did ask us to follow the instruction: VARY, to re-arrange all 25 letters involved.  What follows are my alternative results …

Introduction: I retire at Elap’s masterclass

Describing the squares and their contents: similar aspect (as letters are)!

Describing both letter square constructions: all are artist’s masterpieces!

And describing the endgame.  Crisp tail: same letters, areas

Great fun – thanks again!  In summary: Elap is a secret trial master!!

cheers all,



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Listener No 4477: Word Squares by Elap

Posted by Dave Hennings on 8 Dec 2017

Last year’s Elap problem was about the Collatz conjecture and hailstone numbers. (A Youtube channel that I have recently come across is Numberphile which deals with a whole host of fascinating mathematics — see here for one of its Collatz videos.)

This week, what looked like a fairly interesting set of just 22 clues. Where there was a single algebraic expression, the answer was its square; where there were three expressions, the first was the sum of the squares of the other two and was the answer. Each letter was less than a hundred and the sum of two different non-zero squares.

As I’ve said before, there is normally just one starting point for a mathematical puzzle. Here, it was unlikely to be a clue like 13dn AD + oo, A + C + e – W, O + t – M – T (4) but more likely to be 12ac (which had two clues) D/P (2) and A (3) or 16ac I (2).

Anyway, before going any further, I constructed a little table of the sums of squares less than 100:

1 2 3 4 5 6 7 8 9
1 X 5 10 17 26 37 50 65 82
2 5 X 13 20 29 39 52 67 85
3 10 13 X 25 34 45 58 73 90
4 17 20 25 X 41 52 65 80 97
5 26 29 34 41 X 61 74 89
6 37 40 45 52 61 X 85
7 50 53 58 65 74 85 X
8 65 68 73 80 89 X
9 82 85 90 97 X

It would have probably been easier just to list out the distinct 28 values, but the table did just fine.

Starting with 12, where the first two digits and the whole number were squares, that had to be 169, 256 or 361, with A = 13, 16 or 19. However, only 13 was the sum of two squares (or SOTS as I put in my notes), so one down and 21 to go.

It was fairly near the beginning of the whole process that I remembered a numerical (Arden’s Square-bashing back in 2010, I think) where the correct solution depended on realising that the square root of a number can be positive or negative. I wondered if this would happen here.

I then seem to forget all about that until I solved 2dn A + t – N (3) which was 13 + 68 – 97 giving -16. A short while later, I got to the end of the puzzle anyway so no real negative square issues.

Except, I had both E and a equal to 53!


Luckily, I didn’t have to go right back to the beginning, and found 13 Y, R – C, E – C – P (2) to be the culprit. Changing E from 53 to 37 fixed the problem and I breathed a sigh of relief.

Mind you, the bottom half of each grid looked a bit sparse and presumably the letters in numerical order would help resolve it: they spelt out ILAPCREMSTVarytWoDOZeN. At first I wondered if I’d got the first bit wrong and it should be ELAP….

This could be split more clearly into the three parts required by the preamble: ILAPCREMST for the letters 0–9; Vary — what needed doing to the letters in the first grid to give the second; and Two Dozen giving the total number of 5-letter words in the final grids, which I assumed would be across, down and diagonal, excluding upwards and backwards.

A short while later, after a bit of letter-matching and shuffling, I ended with the required number of words. Some were a bit weird and needed checking in Chambers, especially CEILI, ARERE and TRASS.

About par on the stopwatch for a mathematical for me, and rewarding to get to the end without too much back-tracking! Thanks, Elap.

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Word Squares by Elap

Posted by shirleycurran on 8 Dec 2017

I’ve been dreading the Friday download all week. Yes, this OCD of Listener solving reaches the point where that three-monthly numerical can disrupt our existence for days. When we saw the length of the preamble (several times the number of letters in the clues) and tried to get our heads round what we were being instructed to do, we were truly discombobulated. This was not going to be easy.

Could Elap hope to have his Listener Tippler’s Club membership renewed with such an offering? I searched through his clues and found DeW – that’s one of the world’s top selling whiskies isn’t it? (It was also one of the last clues we solved when we had slogged for about five hours and found that D = 80, e = 89 and W = 73 which gave us 9216 to enter). I wonder whether that PASTS in Grid 1 was a careless spelling of PASTIS. Benefit of the doubt to Elap so “Cheers”. See you with the Pastis in Paris?

The other Numpty soon worked out that of the 34 available digits between 2 and 98 that could be the sum of two squares, 2,8,18,32,72 and 98 were not available as they are all the sums of identical squares, so we were left with 28 potential integers that had to be the equivalents of the letters I,A,P,C,R,M,S,T,E,Y,O,D,Z,L,N,V,W and t,o,e,a,r (there would be six left-overs). That didn’t seem quite so daunting and we set to work with him filling the usual mountain of paper and complaining at my slowness with the calculator.

Initially the grid fill went well but we hit our first brick wall when we found that we had E = 53 and o = 53.  We had O at 89 and t at 61 at this stage and things had been looking good, but it was not to be. I don’t think my O Level maths teacher ever told us that a negative number squares to a positive, but the other Numpty knows that sort of thing and with lots of cursing, we extricated ourselves from our mess which meant rethinking O, t and N among others.

Enough – you wouldn’t be reading this if you hadn’t completed the puzzle. So on to the endgame.

Once our grids were complete (well, all the clues were in) it took five minutes to order the letters by increasing numeric values and we found the ten letters that must replace the digits 0 to 9 – ILAPCREMST, the hint that we had to VARY the positions of the letters in the first grid to arrive at the second grid and the information that we were looking for TWO DOZEN five-letter words.

At this point, I should admit that during our flailings, the word TWENtY had obligingly appeared and totally misled us about how many words were going to appear in the word squares, but TWO DOZEN! That is an achievement in itself and, of course, required two words going diagonally in each grid or some going in two directions (actually I found 26 by counting STIME and EMITS, TRAMS and SMART as well as the four diagonals, with SAIRS being a valid Scottish word but I suppose that is just nit-picking).

Converting the number grids to word grids was almost fun. That’s how I like my crosswords – WORDS! – but we were faced with gaps and were told that we had to VARY the positions of the letters in the first grid to produce the second. Crossword compiler told me that there was only one way to complete grid one and that gave SEERS on the last line and the spare letters T,M,S,S,I,S to use to complete the second grid

Ah, the HARE. He was there in 2d, running round like a headless chicken or burying his head in the sand at the thought of a numerical crossword, and there was a MARA doubling back on himself at the top of the second grid but it was another solver who actually told me how to fill the second grid. I had to anagram or jumble those 25 letters of the first grid and that gave SILLIEST TRAP SECRETES A MARA. Why didn’t I spot that for myself? Simples!

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