Checkers checked...

There are 64 squares in a chess board and there are 9 horizontal and 9 vertical lines. To form rectangles we need 2 horizontal and 2 vertical lines from each set. This can be done in

$^9C_{2} * ^9C_{2} ways$

=36 * 36

=1296 ways

This is my solution: A rectangle can be defined by two diagonally-opposite vertices. An 8x8 rectangle has 81 (9 x 9) lattice points. A rectangle cannot be defined by two points that are in the same row or column, therefore when we choose our first lattice point, it rules out a row and a column of lattice points. There are therefore one row and column less from which we can choose our second point: leaving the next smallest square, which is 64 (8 x 8). However, we are over-counting. Take a rectangle ABCD, where A and C are diagonally opposite. This rectangle will hence be defined by AC, CA, BD & DB. Therefore, we must divide by 4.

$\frac { 81*64 }{ 4 } = 1296$