Rectangles on a Chessboard

Date: 02/09/2002 at 11:10:26
From: Dave
Subject: Number of rectangles on a chessboard

Dear Dr Maths,

How many rectangles are there on a chessboard?

Date: 02/10/2002 at 00:53:06
From: Doctor Schwa
Subject: Re: Number of rectangles on a chessboard

Hi Dave,

Here's a hint on this question that will reveal a beautiful pattern:
start in the top left square of the chessboard, and write in each
square the number of rectangles whose BOTTOM RIGHT corner is at
the bottom right corner of that square.

So in the first square you'll write 1 and in the second square you'll
write 2 (because there are a 1x1 and 1x2 rectangle that will both have
their bottom right corner there) ...

Give that method a try, and see how it goes.

If you get stuck, there are two other methods of solving this problem
in the Dr. Math archives, at

   Number of Squares in an NxN Square
   http://mathforum.org/dr.math/problems/fengh.7.29.96.html   

and a slightly different version of those same two methods is at

   Squares, Rectangles on a Chessboard
   http://mathforum.org/dr.math/problems/taur8.14.97.html   

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/