Let's describe again the structure of the square. We construct it recursively starting with a single cell - a 1x1 square. The next step generates a 2x2 square, the third on - 4x4, and so on. The four pieces may also be designated as S nw , S ne , S se , and S sw . The entry in each cell is the smallest nonnegative integer that has not been yet used to its left in the same row and above it in the same column.
All squares are symmetric with respect to the diagonal from the top left corner.
The entry in each cell is the smallest nonnegative integer not less than 2 N-1 that has not been yet used to its left in the same row and above it in the same column. Which concludes the inductive proof. Moreover, the explicit structure of our Latin square illuminates the relationship between the Mex Rule and the XOR operation applied on the set of all nonnegative integers. Exercises Prove the above Proposition. Alexander Bogomolny 2413159 .