Knights
Logic
Level
2
What is the maximum number of knights in an chessboard, so that no knights can attack each other?
The answer is 32.
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1 solution
1) First, we show that 32 is possible.
Place the knights on the 32 white squares. Then, the squares that such a knight attacks must be black, hence it doesn't attack any other knight that is on a white square.
2) Now, we show that 32 is an upper bound.
Break up the board into 8 smaller boards of size 2 × 4 . In each board, label the squares as such:
On each of these smaller boards, we see that we can place at most 1 knight in each of the pairs of squares with the same number, since these pairs of squares attack each other. Hence, there are at most 4 knights on this small board.
Since we have 8 different small boards, this shows us that we can have at most 8 × 4 = 3 2 knights on the total chessboard.