When he actually counted them and found that he was right, there were 16, he was delighted.
The crucial final point is that the side length of the missing square at its center is the difference between the two unknown numbers.
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The sum of these unknown numbers is known, so its square is as well. And this is 4 times their product, plus the square of their difference.
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So the square of their difference is known, and the difference itself is the square root of its square -- up to an ambiguous overall sign.
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