A square grid of distinct integers is a magic square if all its rows columns and full diagonals have the same sum. Otherwise it is not a magic square.

Now suppose we fill a square grid with samples from a continuous random variable. The probability that the entries are distinct is 1, but the probability that the square is magic is 0.

We could make this more interesting by asking **how close** to magic the square is, using a continuous measure of degree of magic, rather than simply asking whether the square is magic or not.

If we have an *n* by *n* square of real numbers, we could take the set of *n* row sums, *n* column sums, and 2 diagonals as data and look at its range or its variance. These statistics would be zero for a magic square and small for a nearly magic square.

If each element of the square is a sample from a standard normal random variable, what is the distribution of these statistics? How much does the distribution on the numbers matter?