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also, to prove isomorphism, you need to convince yourself that all possible ways of making 15 with 3 numbers are represented in the magic square. Can't think of an elegant way to do that, but if you examine it, you'll see that nothing's been left out.
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Try examining the "order" of each card. Define "order" to be "the number of winning combinations that use this card".
5 has order 4.
2, 6, 4 and 8 have order 3
9, 3, 1 and 7 have order 2
this lines up with the orders of the tic-tac-toe squares. This isn't much different than listing all of the solutions and comparing them in the end, but I think it looks prettier.