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All of Charlie's triples have an even square in them. Using that, the triples for Charlie are (squared=sq):
1sq+0sq=1sq
3sq+4sq=5sq
5sq+12sq=13sq
7sq+24sq=25sq
9sq+40sq=41sq
11sq+60sq=61sq
13sq+84sq=85sq
There is a pattern. The smaller, odd, square number will always rise by two. The larger, even, square number will go up by multiples of four in order, 4,8,12,16,20,24 etc. The answer will always e one square number more than the largest number in the equation. The next equation in the pattern will be
15sq+112sq=113sq. 15sq=225, 112sq=12544, and 113sq=12769. 12544+225=12769. And to prove my method is right, 15=13+2, 112=84+28, and 113=112+1.