## 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

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