Newsgroups: rec.puzzles
From: Dan Hoey <haoyuep@aol.com>
Date: Fri, 06 Jun 2008 17:32:43 -0400
Subject: Re: Partition a square into rectangles

Risto Lankinen wrote:
> no.glamour@gmail.com wrote:
>> Hello,
>> Prove/disprove that a 3000X3000 square can be partitioned into 5X9
>> rectangles (the rectangles can be mixed, either 5X9 or 7X9).

and Risto disproves it.

There are over 30 proofs of the fact that if we tile a rectangle
with rectangular tiles, where each tile has at least one integer
side, then the rectangle will have at least one integer side.
Thus it is impossible to tile a (1000/3)x(1000/3) square with
(5/9)x1 and (7/9)x1 rectangles.

One proof is to color the rectangle in a (1/2)x(1/2)
checkerboard pattern with one corner of the rectangle on the
checkerboard grid.  Each tile will cover the same area of
each color, while the rectangle will not have the same area of
each color unless it has an integer side.

Dan Hoey
haoyuep a t aol.com