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Topic: tiling a defective chess-board
Replies: 1   Last Post: May 20, 2000 10:25 AM

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 Kathleen Posts: 58 Registered: 12/6/04
tiling a defective chess-board
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 "chess-board" that has 2^n rows and 2^n
columns. You are given an endless supply of L-shaped tiles - each one
is a 2x2 chess-board with the upper right square removed. Now,
suppose somebody removes a single, random square from your original
chess-board. Show that you can completely cover the remaining squares
with your L-shaped pieces, without overlapping any of them.

HINT: If you divide your chess-board in half, vertically and
horizontally, you get four smaller versions of the same problem. Can
three of these sub-problems combine to leave a handy L-shaped hole?

I started by drawing out the first few defective chess-boards 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

Date Subject Author
5/18/00 Kathleen
5/20/00 macduff