Question 82
Let $n$ be a positive integer. In this problem, we consider labellings of the squares of a chessboard of size $n \times n$ with the natural numbers from $1$ to $n^2$ such that every number is used exactly once. Given such a labelling, we say a positive integer is a *rook product* if it is the product of the labels of $n$ squares which have the property that if you place a rook on each of them, no two rooks will attack each other.
(Two rooks are attacking each other, if and only if they are in the same row or column.)
a. Let $n = 8$. Determine whether there exists a labelling of an $8 \times 8$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $65$.
b. Let $n = 10$. Determine whether there exists a labelling of a $10 \times 10$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $101$.
(Two rooks are attacking each other, if and only if they are in the same row or column.)
a. Let $n = 8$. Determine whether there exists a labelling of an $8 \times 8$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $65$.
b. Let $n = 10$. Determine whether there exists a labelling of a $10 \times 10$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $101$.