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Let $n\ge 6$ be an even number. An $n$ by $n$ square is tiled by $2$ by $1$ dominoes. They can be placed horizontally or vertically. Must there exist a fault-line, or a line cutting the rectangle without cutting any domino?

I have no clue how to start with this problem. Any hint would be appreciated!

Kai
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  • This is a well-known problem ... so you can just Google around and most likely find the answer. But, if you want to figure it out yourself, I would say: just start tiling! After a while, you'll form some insights, which can turn into hypotheses, and possibly theorems. – Bram28 Sep 11 '19 at 22:21
  • Thank you very much! – Kai Sep 11 '19 at 22:21
  • For all even $n > 6$, fault free tiling of a $n \times n$ square by domino is possible. see ref in my answer to a related question. – achille hui Sep 11 '19 at 22:35

1 Answers1

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There is a nice bit of arithmetic for the case $n=6$.

There are 10 possible fault lines, each of which must cut an even number of dominoes.

There are 36 squares and so there are 18 dominoes, each of which cuts precisely one possible fault line.

Therefore one possible fault line cuts no dominoes i.e. it actually is a fault line.

As $n$ increases, the number of dominoes increases much faster than the number of lines and so you can arrange for dominoes to cut each possible fault line.