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Peaceful knights on a small board

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What is the largest number of knights that can be placed on a $5$ by $5$ board, at most one per cell, so that no knight attacks another? A knight attacks the cells a chess knight can move to, which are the cells at distance $2$ in one direction and $1$ in the other.

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Hint

A knight always moves to a cell of the opposite chessboard colour. For the upper bound, pair up cells that attack each other.

Solution

A knight always moves to a cell of the opposite chessboard colour, so knights on all $13$ cells of the colour of the corners attack nothing, which shows that $13$ is possible.
For the upper bound number rows and columns from $1$ to $5$ ; the $12$ pairs $(1, 2) - (3, 1)$ , $(1, 4) - (2, 2)$ , $(2, 1) - (4, 2)$ , $(2, 3) - (4, 4)$ , $(2, 5) - (1, 3)$ , $(3, 2) - (5, 1)$ , $(3, 4) - (1, 5)$ , $(4, 1) - (5, 3)$ , $(4, 3) - (5, 5)$ , $(4, 5) - (2, 4)$ , $(5, 2) - (3, 3)$ , $(5, 4) - (3, 5)$ are each a knight move apart, and only cell $(1, 1)$ is left unpaired.
From each pair at most one knight can stand, so at most $12 + 1$ knights fit.
Hence the largest number is $13$ .

Answer: $13$

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