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tiling a defective chessboard
Posted:
May 18, 2000 1:00 PM


I hope somebody can help me with this question. I'm kind of stuck ...
The qn is:
Suppose you are given a "chessboard" that has 2^n rows and 2^n columns. You are given an endless supply of Lshaped tiles  each one is a 2x2 chessboard with the upper right square removed. Now, suppose somebody removes a single, random square from your original chessboard. Show that you can completely cover the remaining squares with your Lshaped pieces, without overlapping any of them.
HINT: If you divide your chessboard in half, vertically and horizontally, you get four smaller versions of the same problem. Can three of these subproblems combine to leave a handy Lshaped hole?
I started by drawing out the first few defective chessboards and found that for a 2x2 board you had 1 L shaped piece (triomino) 4x4 > 5 triominos 8x8 > 21 "
that pattern seems to suggest that each board will have 4 times the number of triominos the previous board has plus 1. Then I thought this doesn't help me any with this qn....
Maybe some sort of constructive proof would be the way to go? I don't know & I'm not sure how to get started. Could anyone help please?
Thank you, Kathleen



