Let's Play Some Chess-2

Its clear that a chess board has 32 black squares and 32 white squares. Now, any knight in white square goes to black square in next move, and vice-versa. Now if we place all the knights in white(or black), then they will be in non-attacking position, since their next move will be to a black(or white) square. Thus, the no. of knights is 32. Now, we'll show that its not possible to place more than 32 knights. Let if possible, at least 1 more knight can be placed. Thus, it must be placed in a black square and hence, it will be in an attacking position with some other knight in white. Thus, the max. no. of knights is 32.

Let there be $m$ knights on black squares and $n$ on white squares.

Note that a knight on a white square can only attack the knights on black squares and vice-versa.

So, let the number of black squares that those $n$ knights present on the white squares attack = $a$ .

$\Rightarrow$ $32-a=m$ .

It is easy to prove that $n \leq a$ .

$\Rightarrow$ Total number of knights= $m+n$ = $32-a+n$ $\leq$ $32$ . And it is easy to get $32$ knights by keeping either of $m$ or $n$ = $0$ .

Thus, the maximum number of knights that can be placed on a chess board such that no knight is attacked by any other knight = $32$ .