How many rectangles of any size are on a 8x8 chess board.?

Question

I know that the answer is 1296 (including squares) and that the equation is (1+2+3+4+5+6+7+8)(1+2+3+4+5+6+7+8) but how does this formula work. Explain.

Answer 1

There are several different formulas, not all of which have a "visual" interpretation as you look at a checkerboard.
Familiarize your self with The On-Line Encyclopedia of Integer Sequences:

http://www.research.att.com/~njas/sequences/?q=number+of+orthogonal+rectangles+on+a+checkerboard&sort=0&fmt=0&language=english

For this sequence the calculation is worked out for all size checkerboards starting with 0 x 0. Notice that 1296 is the ninth entry.

Notice that the sequence is named: Sum of first n cubes; or n-th triangular number squared.

Triangular numbers are numbers of points in a triangular array. For example ten is the fourth triangular number because there are ten pins in a normal bowling alley(four rows of pins). The eighth triangular number is 36, which is also (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

Answer 2

ask Carl Sagan ;) He'll also tell you it's a HUGE amount of grain if you get 1 grain for the 1st square, and double each one after.