From - Sat Sep 19 21:31:23 1998
From: ilan@isgtec.com
Newsgroups: sci.math,rec.puzzles
Subject: Re: Russian 33 puzzle
Date: Sat, 19 Sep 1998 15:05:33 GMT
Lines: 79
John Scholes wrote:
> (a) A 6 x 6 board is tiled with 2 x 1 dominoes. Prove that we can always
> divide the board into two rectangles, each of which is tiled separately
> (with no domino crossing the dividing line).
> (b) Is this true for an 8 x 8 board?
>
> John Scholes
>
SPOILER
There are 10 internal lines on the 6x6 board, 5 horizontal and 5 vertical.
There must be a domino placed across the first horizontal line. As this leaves
5 squares between this line and the edge, another domino must be placed across
this line. Similarly two dominoes must be placed across each of the other
lines. As there are 10 lines, 20 dominoes are needed, but only 18 dominoes
can be placed over the board.
On an 8x8 board such a tiling is possible (there are many):
+--+--+--+--+--+--+--+--+
| | | | | |
+--+--+ +--+--+--+--+ +
| | | | | | |
+ + +--+--+--+--+--+--+
| | | | | | |
+--+--+--+--+ + +--+--+
| | | | | | |
+ +--+--+ +--+--+--+--+
| | | | | | | |
+--+--+--+--+ + + + +
| | | | | | | |
+--+--+ + +--+--+--+--+
| | | | | | | |
+ + +--+--+--+--+ + +
| | | | | | |
+--+--+--+--+--+--+--+--+
__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||
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