Prove that the maximum number of knights that can be placed on a chess board NxN such that no knight attacks another is n^2/2

Question

I need to prove that the maximum number of knights that can be placed on a chess board n x n such that no knight attacks another is $n^2/2$ for even n and $(n^2+1)/2$ for odd numbers. I guess that all knights should stand on the squares with the same color. And for example I can look at chess board 2 x 4 and show that the max number of knights is 4, but how can I use this to prove the whole statement for board n x n? Thank you!