# Posts Tagged ‘Equality’

## 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!

Cheers,

Tim / Encota

## 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”