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captaincoma,

a = 53
Y = 65

Let me know if you need any more help.

No need to back-track all the way Dylan, just take another look at 2d. If you take the negative root of the answer you have this will give different values for both t and N, allowing you to recalculate O and progress to the end. There is also a negative expression in one of the longer down clues.

I think I may throw in the towel on this one. I realized from the start that some of the equations might be negative numbers, so took that into account, but I've still ended up with an impasse since I cannot get valid second and third equations for 15a, which means at least one of my values for C, Y, M, P, A must be wrong. I'm certain of A and Y. It's C that is causing the problem either here or elsewhere. It's possible I've gone wrong with D and P since there are several possibilities there.

Solvers might find this site useful to find the summed squares for a given number:

http://nonagon.org/ExLibris/fermat-sum-two-squares-calculator

Thanks demeter. I finally finished the numbers part after considering the negative options.

Has anyone solved the second part? It's a complete mystery to me.

captaincoma,

No, I still don't see what to do. We have to VARY the positions of the letters in the first grid to arrive at the second. What does this mean exactly?

Dryden,

C = 25
D = 80
P = 20

Dryden, thanks for the link. Though I'm hopefully over most of that phase now, I was haunted that I might be overlooking some possibilities. I think after solving Y, there were only 2 possibilities for each of P and D; but still I agree, it was a tortuous process. Perhaps the was another road which I did not see...

Demeter

Same letters in each grid, but in a different arrangement.

Add the four letters that appear in the first grid but not the second to the second and the three from the second not in the first to the first. The original letters don't move and that just leaves the same two letters missing from each grid to complete the word squares.

What a trauma! 3 false starts yesterday, so I gave up. Finally, after 5 hours this morning I got the numeric solution. The end game is very clever, but how one sets about compiling a puzzle like this is beyond my comprehension - bravo Elap.

Thanks, Demeter. Those are the values I have, so perhaps I'll persevere.