Question 67
Consider an operation of putting an integer greater than or equal to $1$ and less than or equal to $6$ to each square in a grid of $6 \times 6$ squares. For a pair of integers $(i, j)$ with $1 \le i, j \le 6$, denote by $i \diamond j$ the integer put into the square located on $i$-th row and $j$-th column after an operation. How many different operations are there which satisfy the following 2 conditions?
* For every $i$ ($1 \le i \le 6$), $i \diamond i = i$ holds.
* For every choice of $i, j, k, \ell$ with $1 \le i, j, k, \ell \le 6$, $(i \diamond j) \diamond (k \diamond \ell) = i \diamond \ell$ holds.
* For every $i$ ($1 \le i \le 6$), $i \diamond i = i$ holds.
* For every choice of $i, j, k, \ell$ with $1 \le i, j, k, \ell \le 6$, $(i \diamond j) \diamond (k \diamond \ell) = i \diamond \ell$ holds.