On an 8x8 chess board we place rooks so that the number of them is odd on each line or collumn. Show that the number of black squares that have rooks is even.
What I found: Obviously, since the sum of all rooks is even, the numbers of black and white squares with rooks have the same parity. Also since every line or column have even number of squares, the number of them with/ without rooks have the same parity so every arrangement generates another one. This is basically the only relevant thing I have got. I still dont know how to get the parity
I have seen that this is a duplicate and I am sorry Here is the solution (I lost the stack exchange link) https://www.quora.com/If-every-row-and-column-on-a-chess-board-has-an-odd-number-of-pieces-on-it-how-can-I-prove-that-there-are-an-even-number-of-pieces-on-black-spots