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Before we find a function to minimize, we should first find some equalities that determine whether a grid $(a_{ij})$ is a magic square. First off, a magic square has the restriction that the entries $a_{ij}$ are some permutation of $1$ to $n^2$. Here's a trick to ensure that: We just need to ensure that the polynomial $(X-a_{11}) \cdot (X-a_{12}) \cdots (X-...


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Suppose there is an $n$ X $n$ square; so clearly it has $n^2$ numbers: from $1$ to $n^2$. Also, let $K$ be the magic constant. Since this is a magic square, the sum of each row, column and diagonal are equal. For our purpose, let's consider the row-sums only. Since there are $n$ rows which contain all the numbers from $1$ to $n^2$ exactly once, the sum of $...


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