The most knights that cannot attack each other

Question

What is the largest number of knights that can be placed on an 8 by 8 chessboard so that no two of them attack each other? Give an arrangement that achieves your number, and prove that no arrangement can do better. Then compare with the corresponding numbers for rooks, bishops and kings.

Accepted Answer

32 knights: put one on every square of a single colour. 32 is achievable. A knight on a light square attacks only dark squares. So 32 knights on all 32 light squares attack no square that holds a knight. None of them attacks another. 32 is the maximum. Split the board into eight 2 by 4 rectangles. Within a 2 by 4 rectangle, the 8 squares can be paired up so that each pair is a knight's move apart: the square in row 1, column 1 pairs with row 2, column 3; row 1 column 2 with row 2 column 4; row 1 column 3 with row 2 column 1; row 1 column 4 with row 2 column 2. So the whole board splits into 3…

The most knights that cannot attack each other

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