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Like others I spent about an hour reading and processing the setup. After understanding that it took about the same time to fill in the grid. Nice one Oyler.

Well I've done it, including the endgame, and I think I've done it right.

But then I re-read in the first para of the preamble, that it's "based on sets of 3 district positive integers where the sum of any two or all three is a triangular number". That certainly doesn't apply to my triples...in fact I don't see how such a situation could exist.

Am I being stupid here?

Steve

Steve - Each equation uses 3 distinct integers in that way - haven’t you used that fact to solve the equations and fill the grid

Hi Candledave

Yes I've used 3 integers in the LHS of each equation, but they mostly don't have the property that any pair chosen from the 3 sums to a triangular number

Steve, the integers on the LHS are derived from the triangular numbers which themselves are sums of the integer triplets. After solving the equations you have to work backwards to obtain the original triplets. While this isn't needed for filling the grid, it's necessary for finding the thematic sum.

I hugely appreciate your attempts to help me understand, cypherhouse (& candledave), but I'm still not getting it.

I do have a thematic sum, which worked out as a three-digit number with a suitably thematic property. Does that sound right?

And when you say I have to find the "original triples", do they have the property that any 2 will add up to a triangular number?

Brain hurts. (And yes, I also have a maths degree 🙄)

Thanks, Steve

steve - you have presumably found p, q, r (and s). P, q and r are the roots of triangular numbers and you have to find the sets of three integers so that 2 of them each add together to form the p, q and r related triangular numbers and all three sum to s

Is that any clearer?

Does anybody else find it ironic that the default prize for a numerical is still a dictionary of words? Any suggestions for something more appropriate? Perhaps [/i The Encyclopedia of Integer Sequences] by N. J.A. Sloane and Simon Plouffe? I know it only contains a [/b tiny] fraction of the sequences in the on-line edition, but there's also some interesting text.

(*note ABC below are different from ABC in the puzzle. Apologies for the poor notation, this is just what I used)

I used this notation. A, B, C are the three integers that sum to triangle numbers.

A+B = p(p+1)/2
A+C = q(q+1)/2
B+C = r(r+1)/2
A+B+C = s

With the clues solved and the grid filled you should have p,q,r, and s for each equation. So it's just solving the above for A, B, and C

Many thanks cypherhouse!! 👍