Rectangles

Choose $2$ of the $9$ horizontal lines on the board, (edge to edge), and $2$ of the $9$ vertical lines on the board, (again edge to edge).

The region within the $4$ points of intersection of these $2$ pairs of lines is, of course, a rectangle, and is in fact unique among the rectangles that can be formed in this way. So we have a one-to-one correspondence between the rectangles on a chessboard and the $\binom{9}{2} * \binom{9}{2}$ possible pairs of parallel lines as outlined above.

Thus the number of rectangles is $\binom{9}{2} * \binom{9}{2} = 36 * 36 = \boxed{1296}$ .

In n × n grid, number squares = 1² + 2²+ 3² + ... + n² = n(n+1)(2n+1)/6

In n × n grid, number rectangles = 1³ + 2³+ 3³ + ... + n³ =[ n(n+1)/2]²

8 × 8 grid number of rectangles = [8 × 9/2]² =1296

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 largest 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.

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