Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?
Problem
Source: MOP 2006 Homework - Red Group #6
Tags: geometry, rectangle, geometric transformation, rotation, combinatorics unsolved, combinatorics
chaotic_iak
10.06.2014 11:00
What is peering through the tiling?
BOGTRO
11.06.2014 09:01
I'm guessing "using a needle to peer through the tiling" means that there exists a line parallel to the sides that does not intersect the interior of any domino.
Zimbalono
18.06.2014 16:58
There are two solutions for the $5\times6$ rectangle, six if you count rotations and reflections.
[asy][asy]for(int i=0;i<7;++i){draw((0,i)--(6,i)^^(i,0)--(i,6));if(i<5)label(string(i+1),(i+1,6),N);}[/asy][/asy]
At least one domino must straddle the $i$th line. But then there an odd number of remaining squares on either side of the line, so another domino must straddle the same line. Therefore there are at least ten horizontal dominoes. By the same argument, there are at least ten vertical dominoes. But there isn't enough room for twenty dominoes.