The Knights Dilemma

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We can divide the chessboard into 32 groups each with two squares(I only coloured 8 groups, you can colour the rest using the same pattern.) such that the squares are a knight's move away.

Let the number of knights that can be placed be $n$ . If $n > 32$ , then by PHP, two knights must be in the same group. But then they will attack each other. Therefore $n \leq 32$ . To prove that $n = 32$ is the maximum, we must find a configuration of 32 knights.

Note that a knight always attacks squares of a different color.(i.e. if its on a black square, it attacks only white squares.) Therefore, if we put all the knights on squares of the same color(WLOG black), then they will not be able to attack each other. Total number of black squares = 32.

Therefore the maximum number of knights which can be placed on a 8X8 chessboard so that they do not attack each other is $\boxed{32}$ .

If you play chess you would know that a knight can only attack or move to a square of the opposite color as the square where it came from (so it's always Black -> White and White -> Black). If you put all knights on the same square color then they can't attack each other. Since in a 8x8 board there are 32 white squares and 32 black squares, then the maximum number of knights is 32

Simply the knight must move to the other color, if it was on white it can't attack white, so you have 32 white squares you can place them all.

The knights always attack each other only on squares of opposite colours. so place all the knights on the squares of the same colour. there are 32 squares of the same colour on the chessboard. So the answer is 32

A knight , if placed on white square can only attack on a black one OR

if placed on a white square can only attack black square

so, maximum number of knights which can b placed on a 8 X 8 chessboard would either be the number of black squares or the number of white squares( becauz if not to attack each other, they have to lie in the same color) therefore, the answer is (8 X 8)/2 (i.e. the number of black/white squares)

4 in each row..one in every 2 cells of a row...