Can you tile a 6x6 chessboard with dominoes, without overlaps or gaps, so that the number of dominoes oriented horizontally is equal to the number of dominoes oriented vertically? Why or why not?
$\begingroup$
$\endgroup$
asked Jun 6, 2015 at 5:11
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
The answer is
No
Proof/construction:
Paint the board with a horizontal stripe covering every other row, which colors 18 cells. In a tiling as described, each of the 9 vertical dominoes covers one colored cell, and each of the 9 horizontal ones covers an even number of colored cells (0 or 2). But that means an odd number of colored cells are covered, which can't be 18.
-
$\begingroup$ Beautiful. BTW, I can across this problem while trying to form a proof to this question. $\endgroup$ Commented Jun 6, 2015 at 6:08