How Many Rectangles?
This problem was provided by Michael Khoury, US Math Olympiad Team Member.
How many rectangles (of any size) are present on a standard 8×8 chessboard?
Correct solutions were submitted by: Dakotah Martinez, Bethany Witemeyer
Choosing a rectangle is equivalent to choosing the four lines that bound it. There are nine lines running in each direction so there are 9C2 = 36 ways to choose the horizontal borders. (nCr is the binomial coefficient, out of n choose r where order of selection does not matter.) Similarly there are 36 ways to choose the vertical borders. The total number of rectangles is therefore 362 = 1296.
