From Ferguson's Game Theory:
A Latin square is an $n \times n$ array of $n$ different letters such that each letter occurs once and only once in each row and each column. Such games have simple solutions. The value is the average of the numbers in a row, and the strategy that chooses each pure strategy with equal probability 1/n is optimal for both players. The reason is not very deep. The conditions for optimality are satisfied.
I can see that choosing each pure strategy with equal probability satisfies optimality conditions, but
why is this so is there some property of the latin square that allows me to solve for this in the first place?
Is it just because the rows and the columns sum to the same so we suspect the uniform distribution is optimal?