Mundia also found the number of rectangles of size (a,b) for each a,b at most n.  Correct answers were supplied by Joe Podwol, Nathan Pauli, Ray Kremer.  Further solutions were submitted by Jens Voss, Jogmin Baek, Ken Keating, Khanh Ngo, Philippe Fondanaiche, Andrew Kapust, Aaron David Kahn, Dylan Covert, Cyril Terakopiantz, Ivan Lisac, Fransesc Suñol, Edward Lee, Aashish Parikh, Ken Buch, Randy Pohl, Burkhart Venzke, Shekhar Joglekar.

This problem brought the most varied number of solutions.  Here is perhaps the simplest:  Place a coordinate system on the checkerboard with (0,0) at one corner and (8,8) at the other.  To choose a rectangle, you need to identify two of the vertices which sit diagonally opposite one other.  Such a pair can be identified by their two distinct x-coordiates and the two distinct y-coordinates; for example, the x-coordinates 1,4 and the y-coordinates 3,7 corresponds to the rectangle whose opposite corners sit at (1,3) and (4,7).  From the nine possible  x-coordinates -- that is from 0 to 8 -- the number of ways to choose two distinct ones is 9!/2!7! = 36, with the same for the y-coordinates.  The total comes to 36² = 1296.

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Page last updated 8 May 2000.
ã2000 Alberto L. Delgado