Question 3
We define a chessboard polygon to be a simple polygon whose sides are
situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers.
These lines divide the interior into unit squares, which are shaded alternately
gray and white so that adjacent squares have different colors.
To tile a chessboard polygon by dominoes is to
exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles.
Finally, a tasteful tiling is one which avoids the
two configurations of dominoes and colors shown on the left below.
Two tilings of a $3 \times 4$ rectangle are shown;
the first one is tasteful, while the second is not,
due to the vertical dominoes in the upper right corner.
Prove that (a) if a chessboard polygon can be tiled by dominoes,
then it can be done so tastefully, and (b) such a tasteful tiling is unique.
Proposed by Sam Vandervelde (credit in the supplied third-party notes).
situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers.
These lines divide the interior into unit squares, which are shaded alternately
gray and white so that adjacent squares have different colors.
To tile a chessboard polygon by dominoes is to
exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles.
Finally, a tasteful tiling is one which avoids the
two configurations of dominoes and colors shown on the left below.
Two tilings of a $3 \times 4$ rectangle are shown;
the first one is tasteful, while the second is not,
due to the vertical dominoes in the upper right corner.
Prove that (a) if a chessboard polygon can be tiled by dominoes,
then it can be done so tastefully, and (b) such a tasteful tiling is unique.
Proposed by Sam Vandervelde (credit in the supplied third-party notes).