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How can I find all the different possibilities for the sum of rows/columns/diagonals for a $4\times4$ magic square? I am not sure how to proceed, except by brute force, which is entirely inelegant.

I should mention that I am looking at a $4\times 4$ magic square with symbols from $0$ to $15$ inclusive.

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    @DanielleHuang Count the same thing two different ways: if the row sum is $r$, then the sum of all the entries in the square must be the sum of row 1, row 2, row 3, and row 4: $r+r+r+r=4r$. Meanwhile, if the square contains each of the numbers from 0 through 15 (or 1 through 16) exactly once, then the sum of all the entries is...2012-07-18

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There is only 1 possible sum : What are the possible sums of an $n \times n$ magic square? That sum would be 30 in a 4x4 magic square with numbers 0-15

EX:

| 0 | 7 |11|12|

|13|10| 6 | 1 |

|14| 9 | 5 | 2 |

| 3 | 4 | 8 |15|