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Posts Tagged ‘Elap’

L4595: Equality by Elap

Posted by Encota on 13 Mar 2020

I found this one tricky – but that was probably me. A couple of silly late-night errors on Friday saw me starting again Sat lunchtime, with it all sorted by 4pm. And when I say silly, I mean ‘putting the answer to 16d in 16a, that sort of stupidity’!

Elap describes the nine sets of 6 numbers as having a remarkable relationship, and I rather tend to agree. A bit of after-the-event Googling took me to something known as the Prouhet-Tarry-Escott problem, which seemed to focus upon two sets of numbers having this property. And from the references in some of these papers, I spotted that at least one other Listener setter appears to be fairly knowledgeable on this subject area …

Like in some mathematical proofs it can be simple to make errors with the edge conditions. One’s favourite online maths site (well, mine anyway) provides the appropriate handling of larger integers, letting me check that 5th powers summed correctly but 6th powers did not. So N’s maximum value is 5. What about the lower end?

It’d be easy to forget that anything to the power zero is 1. I do hope no solvers fell into this trap. So N=0 works, with each set summing to 6. And yes, zero is an integer, if anyone is asking about definitions 🙂

Now double-check that N=-1 does NOT work. A quick calculation shows that the sum of the reciprocals does not work for the first two sets.

So N = 0 to 5 goes in under the Puzzle.

There must be some literature out there for this special set of sets where the numbers are of the form {a, b, c, k-a, k-b, k-c} but I couldn’t immediately find any. I can see they guarantee their sums for N=0 (where the sum is 6) and N=1 (where the sum is 3k) for any set of a, b, c. After that it gets trickier! If there are mathematicians reading this who can provide a gentle pointer for me to relevant background material then I’d be keen to receive it!


Tim / Encota


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Equality by Elap

Posted by shirleycurran on 13 Mar 2020

Years ago, when Chris Lancaster (now the puzzle pages editor for the Telegraph) asked for more Listen With Others contributions from solvers, we offered, but commented that they would sometimes have to be ‘fail’ contributions as we were often unable to solve the ***** things. “No problem” he responded. The Numpties have produced a Listener blog every week since then and planning and writing them has led to a few ‘all correct’ years and just a few errors along the way – and always a completed puzzle. But there are two of us and although the usual ‘Chalicea/Curmudgeon Numpty’ sets the occasional numerical puzzle, despite a relatively good ‘O level’ pass in maths, she can’t solve them for toffee, so, after a groan, handed this over to the in-house Joe Sixpack Numpty – here he is:

Numericals. A dreaded phrase about a remarkable relationship, presumably delightful to a mathmo but tooth gnashing to Joe Sixpack…..

Hmmmmm. This one did look approachable to some extent, some letters being very high powers of integers, especially if you include seeing the “OO” at 3d. This gave good bets for P, D, H, F and O (81, 256, 64, 32 and 27) but then things looked stickier. But what about the helpful message Elap has provided by writing the clue letters in numerical order? Only 23, which can’t form the words ‘SHADE, HIGHLIGHT, LETTERS’,  and so on of the usual Listener puzzles and can’t form ‘DIVIDE, MULTIPLY, ADD’ either. But POWERS is possible, and NTH, and OF! What about SUM OF NTH PoWERs IDentiCAL?  Looks good, and fairly appropriate for the puzzle apart from some niggling doubt about where to use S or s and so on.

Putting the letters in this order and using the ones we already have good bets for, this works out well and allows a grid fill. Finding the upper limit for N to write below the grid required using the boss’s calculator, though, as mine is neanderthal……

Much kinder than it looked at first glance, so thanks, Elap. However did you create it?

Postscript: what about Elap’s remaining in he Listener Setters’ Oenophile Outfit? It was the Sixpack Numpty who pointed out that the grid is overflowing with 69s in various directions. Vat 69 will pass muster, so “Cheers, Elap1”

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Listener No 4569: Bearskin by Elap

Posted by Dave Hennings on 13 Sep 2019

As usual with an Elap mathematical, I get a feeling of dread. That said, all mathematicals fill me with varying degrees of dread. Elap’s last gave us two 5×5 word squares. This time, we had four 4×4 squares, and it seemed they would be wordy as well, given that the digits had to be replaced “preferably all in upper case”.

The letters in clues stood for 27 different integers formed by taking a perfect cube up to five digits and truncating the first digit and any remaining leading zeros. In fact, they had to be “shaved off” and that, together with the title and 13ac’s r-A-Z-O+r would undoubtedly mean something!

As it turned out, this wasn’t too tricky a puzzle. Listing all the 5-digit primes, minus their first digits and zeros, gave 42 such numbers ranging from [2]7 (3³) through to [9]7336 (46³). After truncation, they were from [6]4 to [2]9791.

The starting point was 4through where UUU (3) was a 3-digit number and had to be 7³ = 343. 17dn (E – pp (3)) came next, with p = 4 and E = 331 or 375. Next was 8th (Uz – pp (3)) with a couple of options for z followed by 12ac (CU – UU (5)) where C had three options but, crossing with 17dn, gave 6656 as its value.

From there on, progress was fairly steady although not as quick as I initially thought it would be. It was nice to just have to rely on pencil, paper and a calculator. No doubt some out there decided that a program written in C#+ would be a good way to tackle it!

Sorting the values into numerical order, we ended up with pUzzLiNG (solvers’ activity in filling the grid), dEPIlaTORS (what replaced values 0–9) and A WorD CuBe (what solvers should end up with). The word cube that resulted had 4-letter words running across, down and through the cube.

Thanks for an enjoyable and easy-going mathematical, Elap.

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Bearskin by Elap

Posted by shirleycurran on 13 Sep 2019

Bearskin indeed! We really dislike the three-monthly numerical Listener puzzles so there was plenty of bear-like growling and gnashing of teeth from the other Numpty while I discreetly got out of the way and converted a massive bag of greengages to jam then prepared the supper. Yes, of course I had already hunted for traces of alcohol in those verbal clues and had discovered only a dubious B + ir – i – r. Pretty second-rate beer but cheers anyway, Elap.

The other Numpty’s comment was “Why so many letters in the clues, and why do we have some in both upper and lower case – in particular what was the need for a lower-case l that meant that it had to be specifically distinguished, in the pre-ramble (yes it was a bit of a ramble wasn’t it?) from upper case I?” and “That final clue to ‘pz + Z + L (3)’ must be a gentle joke – or does it have a meaning?”

Clearly it did. After solving U and C the growling began and lasted for hours.

It is only in retrospect that I have wondered how Elap managed to create this puzzle and surmised that he must have struggled initially to create four word squares using only ten letters that could be converted to the digits 0 to 9 (and leaving letters to give ‘A WORD CUBE’ and ‘PUZZLING’) without using any letter more than twice. No wonder he needed that preambular clarification of the lower case l.

With a table of cubes of integers to 100 in Adrian Jenkins’ Number File (you can obtain second-hand copies on the Internet) we constructed a list of potential digits and were ‘puzzling’ from then on for rather a long time, but it was the guess that the first word (‘Sorted by numeric value’) was pUZzLiNG that finally got the Numpty Bear moving forward dramatically rather than sharpening claws (and pencils).

The ‘throughs’ helped us by providing intersecting digits and after many hours of working from letters we had to letters we needed, three of the four layers were completed but we could see no way to produce numeric equivalents for d O u A R and P and fill that bottom layer. It was well after midnight that we decided to place the letters we had in numeric order and that produced:


Penny drop moment at last. The final three words were A WORD CUBE and we could place the remaining d P and O to give DEPILATORS. Ha! So that explained the “Bearskin” heard as BARE SKIN.

Finally there was a smile from the Numpty Bear and as it was by now well after midnight, we jointly converted those nine digits to DEPILATORS and produced our WORD CUBE. What was most astonishing was that it didn’t just have real words on all four layers, going aross and down, but also that the through words were all real too. What a fine and challenging compilation.

(Did I say that? I hate the things and am breathing a sigh of relief that it’s three months until the next one and last one of the Listener year.)


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‘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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