Consider: If we are given a reasonably well-behaved statistical population of real numbers, then samples that are not small will have mean approximately equal to that of the population, right? Then: let n be a large positive integer, and choose n positive integers at random, without replacement, from among the integers from 1 to n squared. Do this a total of n times. (The n squared numbers are depleted at the end of this process, no two selections containing common members.) Let these be the rows of a matrix. The all the rows, columns, and the two diagonals constitute a random sample of size n, and will have mean approximately that of the mean of the integers from 1 to n squared. Therefore this matrix, generated so easily, is already nearly a magic square. It is plausible that all it needs is some tweaking (swapping a few entries here and there) to be exactly a perfect square. Thus, magic squares are, contrary to one’s naïve initial reaction, inevitable, and, indeed, abundant, rather than unlikely or rare. The only surprise is that non-trivial magic squares of very low order exit.
So, here is my first question: In all that I have ever seen about magic squares, this statistical perspective has never been given. But perhaps there is mention that I simply haven’t noticed. That is why I made one of the tags for this question “reference-request”. If I’ve overlooked it, please tell me.
My main question is this: Is it indeed the case that, contrary to first blush, one can plausibly assert, in light of the statistical argument given above, without the need of exhibiting any magic square at all, that magic squares are inevitable and exist in abundance?