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Listener No 4477, Word Squares: A Setter’s Blog by Elap

Posted by Listen With Others on 10 December 2017

The Theme

I started by creating a file containing all of the five-letter words from TEA 2.11. There were 12,125 words including proper nouns.

Out of curiosity I wrote a program to see how many ways ten of these words could be fitted into a 5×5 grid. It took three hours to find that there were 1,152,602,603 different filled grids (excluding reflections).

I then decided to see how many grids there were where the two diagonals were also words, and the answer was 575,560. This took 163 seconds.

I was unhappy with the number of unacceptable words in these, and so I produced a set of 10,508 words which did not contain any proper nouns. There would obviously be other unacceptable words, but it was easier to ignore grids which contained these than to go through the list deleting them (it was easy to remove the proper nouns because they began with a capital letter).

I reran my programs. There were now 229,858,145 grids (the run took 70 minutes) and 115,265 with the diagonals too (taking 71 seconds).

Since there were so many ways of fitting words into the grid, and since I wanted my final grid to be special in some way, it struck me that there was scope to see whether two of the grids could be anagrams of each other.

I decided that the grids should contain ten or fewer different letters so that they could be replaced by digits, thus forming the basis for a mathematical puzzle.

For each filled grid, I wrote the 25 letters (in alphabetical order), followed by the words in the grid, to a file. I then wrote a wee program which sorted the records into ascending order and which then scanned the file to see whether two adjacent records had the first 25 characters the same.

It turned out that there were six possible pairs of grids. Two of them contained similar words somewhere and three contained mildly rude words, but there was one pair which was acceptable. This was very satisfying.

Nature of the Clues

Since I was dealing with word squares, it seemed appropriate to use squares in the clues. The clues would consist of expressions in terms of letters whose values had to be deduced.

I wrote a program which determined which substitution of digits would produce the largest number of perfect squares in the grids. The idea was that some clue values could be squares or square roots – but how would the other clues work? After a lot of dithering, I decided to have some clue values as square roots of the entries, and some consisting of the required answer which was equal to the sum of the squares of two other values. The advantage of this is that the roots could be negative, which might fool some solvers (it fooled me too, because the first vetter spotted that I had two letters round the wrong way in one of the clues which led to a negative answer!).

In line with the theme of two squares, I decided that appropriate letter values could be numbers which could be expressed as the sum of two different squares.

The Hint

I needed a hint, though, which would appear when the letters were sorted by their values. Part of the hint would be the ten letters, in order, by which the digits 0 to 9 were to be replaced. Another part needed to indicate that the grids were anagrams of each other. But how? – there were not enough letters to convey the message. I then hit upon the idea of using lower- and upper-case letters in the clues so that a better hint could be constructed.

The ten letters, represented by 0 to 9, were ILAPCREMST. For the anagramming hint, I first thought of BYJUMBLING to indicate how the second grid was to be derived from the first, but I had also to indicate that the diagonals were words too. I was quickly using too many letters for the clues to be solvable without there being too many of them, and anyway I didn’t want an L in lower case because it would look like the digit 1.

I needed to say that there were TWENTY-FOUR words in the grids, but with ILAPCREMST that used three Ts already! Wait! What about TWO DOZEN?

The hint could then be ILAPCREMST……TWO DOZEN…… with some of the letters in lower case.

Some solvers, though, would assume that there were twenty words in the grids, and so a Y was needed to reinforce this. The word which contained the Y would have to indicate that the grids were anagrams of each other. Hmmm…

What have we got so far?

……ILAPCREMST……TWO DOZEN……

What word, containing Y, could indicate an anagram? If the word had more than, say, six letters we would again be heading for too many letter values for comfort. I needed a short word.

It was a few days later that VARY suddenly sprang to mind. It could be the positions of the letters in the first grid that would have to VARY to arrive at the second grid.

Our hint was now looking like one of these (some of the letters will be in lower case):

VARY ILAPCREMST TWODOZEN
VARY TWODOZEN ILAPCREMST
ILAPCREMST VARY TWODOZEN
ILAPCREMST TWODOZEN VARY
TWODOZEN VARY ILAPCREMST
TWODOZEN ILAPCREMST VARY

The chortle that my wife heard from the study was when I looked at the third option, ILAPCREMST VARY TWODOZEN, and a mean streak in me began to surface.

If there were at least four available letter values between the first T and the Y, wouldn’t some solvers assume that they were the first and last letters of TWENTY? Tee hee!

I changed the case of one of each of the duplicated letters to arrive at this hint:

I L A P C R E M S T V a r Y t W o D O Z e N

The trap was about to be set.

The Clues

I won’t go into details of the derivation of the clues (mainly because I have forgotten now), but the letter values were these:

I L A P C R E M S T V a r Y t W o D O Z e N
5 10 13 20 25 26 37 40 41 50 52 53 58 65 68 73 74 80 82 85 89 97

 
Intentionally, amongst the first values deducible were T = 50 and Y = 65, encouraging some solvers to jump to this conclusion:

  e
…… T W E N t Y ……
50 52 53 58 61 65

 
If this assumption is made, the values of e, L and O would be incorrect, as well as the values of W, E, N and t.

Solvers would most likely end up with these incorrect values:

Letter Correct
Value
Incorrect
Value
E 37 53
e 89 68
L 10 17
N 97 58
O 82 89
t 68 61
W 73 52

 
All the clues except for 7ac and 18ac (the most complex, and likely to be solved relatively late) would still work. Whether or not the correct or incorrect values are used, we have:

In 5ac, 13dn and 14dn O + t is 150
In 6ac and 10ac O – L is 72
In 6ac |N + R – C – e| is 9
In 15ac L + t is 78
In 19ac L + L + t – O is 6
 
In 2dn |A + t – N| is 16
In 3dn and 13dn e – W is 16
In 13dn |E – C – P| is 8
In 4dn |C + W – e| is 9

 
As a retired programmer, one of the lessons I learned early in my career was not to unnecessarily assume anything, and maybe this is a lesson to some of the solvers!
 

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‘Word Squares’ by Elap

Posted by Encota on 8 December 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,

Tim/Encota

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Listener No 4477: Word Squares by Elap

Posted by Dave Hennings on 8 December 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!

Drat!

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 December 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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Listener No 4476, His: A Setter’s Blog by Nebuchadnezzar

Posted by Listen With Others on 3 December 2017

This crossword all started after a long day’s travelling to Croatia, and I remember trying to merge ‘Thinking Outside The Box’ with the 9 dots on a piece of scrap paper. It was approached with the mindset of a doodler, as opposed to “right, I’m going to sit down and write a Listener Crossword”, accompanied by a loud cracking of the knuckles. It was only when I realised the TH could do double-duty that I suspected this was a goer (although, in the interests of my relationship, I postponed the gridfill until my partner and I returned from our holiday).

The puzzle-within-a-puzzle theme is a well-trodden path, but I wanted to pay tribute to what I think is a particularly ingenious challenge of one’s lateral thinking. I was also careful to find a way of citing my references, hence DUDENEY and LOYD down the side — I didn’t want anyone thinking I was trying to pass this off as my own work! I was pleased that there wasn’t too much head-scratching in actually cracking Dudeney/Loyd’s puzzle, I think — I didn’t want to give those who already knew the solution too much of an advantage.

Finally, I needed to create a box outside of which one could think. A bit of research led me to the EGG OF COLUMBUS, and that old ‘letters omitted from wordplay’ trope enabled the square to complete what I felt was quite an aesthetically pleasing final grid.

The gridfill came next, which was achieved using Quinapalus’s magnificent Qxw software. In retrospect, I would have preferred fewer 3/4 letter words with wordplay leading to 2/3 letters, but I was generally pleased with the unching and average length. For the record, the ambiguity of the clue for BULB was recognised by myself, testers and editors prior to publication. I apologise to anyone who felt it affected the overall quality of the puzzle, but I’m of the opinion that, since the endgame removed any ambiguity, all should be forgiven? Perhaps I’m opening a can of worms — please feel free to argue against in the comments!

With my apologies, I don’t remember a great deal of the cluing process. My rule of thumb with these matters is to find a ‘hook’, around which I can build the rest of the clue. To use 25a as a case in point, my mindset will have been something along the lines of: enderon = skin; hen and heron are birds… birds and skin… ah, there’s some sort of surface there about dirty magazines! Then fill in the gaps, whilst trying not to be too clunky when fitting it all together. Those who were kind enough to test the puzzle, and of course the editors, were merciless with some clues. Any that I tried to ‘get away with’ – usually on the grounds that the surface made good sense, but the wordplay was unsound — were rooted out without hesitation, and I’m very grateful for that!

Last and very much least in the process was the title (His = •••• •• ••• in Morse = 9 dots), probably the least satisfying aspect of the puzzle. In my defence, I did feel as though a theme like this justified something that required a bit of lateral thinking. However, I soon realised it was perhaps a tad unfair, and I can only apologise if it detracted from the satisfaction of completing the puzzle. A small part of me still quite likes it, but then again I never had to figure it out.

Finally, I owe a huge debt of gratitude to everyone who contributed to the puzzle’s testing/editing. I know everyone says it in these blogs, but the level of expertise on display when it comes to proofing a Listener crossword is remarkable. I’ll stop gushing now for fear of appearing sycophantic, but I could go on! 
 

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