Question 146
Let $n$ be a positive integer. The integers $1, 2, 3, \ldots, n^{2}$ are to be written in the cells of an $n \times n$ board such that each integer is written in exactly one cell and each cell contains exactly one integer. For every integer $d$ with $d \mid n$, the $d$-division of the board is the division of the board into $(n / d)^{2}$ nonoverlapping sub-boards, each of size $d \times d$, such that each cell is contained in exactly one $d \times d$ sub-board.
We say that $n$ is a cool number if the integers can be written on the $n \times n$ board such that, for each integer $d$ with $d \mid n$ and $1 \lt d \lt n$, in the $d$-division of the board, the sum of the integers written in each $d \times d$ sub-board is not a multiple of $d$.
Determine all even cool numbers.
We say that $n$ is a cool number if the integers can be written on the $n \times n$ board such that, for each integer $d$ with $d \mid n$ and $1 \lt d \lt n$, in the $d$-division of the board, the sum of the integers written in each $d \times d$ sub-board is not a multiple of $d$.
Determine all even cool numbers.