I found the following question online (it was a past question on the BMO):
find all positive integer solutions $x,y,z$ solving the simultaneous equations
$ x + y - z = 12 \ $ and $\ x^2 + y^2 - z^2 = 12 $ .
Normally these questions have a simple and elegant solution and although I've managed to solve it, the answer I found was neither! I wondered if anyone can see a simpler way to tackle the problem.
The way I solved this was to use the first equation to eliminate $z$ in the second question. I then factorised the resulting equation to obtain:
$(x-12)(y-12) = 66$ .
Then, using the fact that $66=2\times 3\times 11$ and also the fact that the equations are symmetric in $x$ and $y$, I worked out the possible factorisations of $66$ into two factors. From this I worked out possible solutions and then checked to see that they indeed were. (In total I found $8$ solutions.) Although this method worked, it seemed a bit cumbersome and I hope someone can improve on it! Please note that this question is aimed at someone with at most A-level standard maths abilities meaning it shouldn't need complex mathematics to solve it!