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Ok, now I subtract points from myself for agreeing to a fake solution. When you choose to traverse along an integer path, you are not necessarily guaranteed to land at a "corner." It is necessary to be at a corner before you are guaranteed to have an integer path along which to travel.
Try again.
[/ QUOTE ] wait a minute, yes you are guaranteed to land at the corner point with at least one other R_i, but i am not sure i can prove that the path necessarily hits the other side.
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This is part of one of the relatively well-known proofs for this problem, which was discussed in Stan Wagon's award-winning paper,
"Fourteen Proofs Of a Result About Tiling a Rectangle" American Mathematical Monthly, Aug/Sep 1987. There is something left to prove. In the spirit of this thread, though, I won't post the rest of the proof in the next few days.