3rd Irish 1990 Problem 1 Find the number of rectangles with sides parallel to the axes whose vertices are all of the form (a, b) with a and b integers such that 0 ≤ a, b ≤ n.
n 2 (n+1) 2 /4 Solution We pick two diagonally opposite vertices. The first may be chosen in (n+1) 2 ways. The second must not lie in the same row or column so may be chosen in n 2 ways. This counts each rectangle ABCD in 4 ways: A,C; C,A; B,D and D,B. 3rd Irish 1990 © John Scholes jscholes@kalva.demon.co.uk 17 Dec 2003 Last updated/corrected 17 Dec 03