We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size
a) $3 \times 3$ exist?
b) $4 \times 4$ exist?
c) $5 \times 5$ exist?
Two tables are different if they differ in at least one cell.
Mark with an X every cell that contains an odd number we now ask for the number of ways to put Xs on the board such that the sum of the number of non X squares in each column and row is even regardless of what column and row we pick.
To put it simpler we want all columns and rows to have an odd number of non X squares or all of them to have an even number of non X squares.
Even more simple we want the number of Xs in each row and column to be either all even or all odd.
The rest seems to be just some simple work so I leave it to the reader to finish.
If it is more then that let me know and I will try to finish it