Here's a possible solution to the puzzle:

Proof of maximalityEdit
Pair up the squares of the board, demonstrated by the pairs of chess pieces on the board below.

Continue this pairing onto the rest of the board. On each pair of such squares, only one knight may be located. Since there are 32 such pairs, it's impossible to place more than 32 knights on the board. Since placing 32 knights is possible, 32 is the maximum number of knights that can be placed on a chessboard so no two attack each other.