Can you tile a heart with dominoes? For a positive integer $n$, let $R_n$ be the set of integer lattice points $(x, y)$ such that

*

*$0 \leq x < 2n$

*$0 \leq y < 4n$

*$x \leq y$

*$y \leq 5n - x$

*$y \leq x + 3n$,

and let $L_n = \{(-x, y) \mid (x, y) \in R_n\}$. Define the $n$th heart board to be $L_n \cup R_n$. The first four heart boards look like this:

Can heart boards ever be tiled with dominoes?
The $n$th heart board seems to have size $10n^2 + 3n - 3$. This is even iff $n$ is odd, so only the odd heart boards have any hope to be tiled. With a computer search I've verified that no heart board with $n \leq 20$ can be tiled using dominoes. I don't see how to extend any "forcing" argument to arbitrary integers, but I suspect that there is some nice pattern. (Any possible tiling seems to only use "upright" dominoes, for example.)
 A: No heart boards can be tiled with dominoes.
Gregory Puleo's suggestion turns into an excellent proof.
"Color" the heart with $+1$ and $-1$ so that the spaces alternate signs. Every domino then covers exactly one $+1$ and one $-1$, so the heart board will not be tileable if there are more $+1$'s than $-1$'s, or vice versa. It suffices to consider the heart a single column at a time.
Fix some $x \in \{0, 1, 2, \dots, 2n - 1\}$. What is the height of the $x$th column? Using the given constraints, it is exactly
$$\min(5n - x, x + 3n, 4n) - x + \delta,$$
where $\delta = 0$ if the minimum is $4n$, and $1$ otherwise. In the first case for the minimum, we get
$$5n - x - x + 1 = 5n - 2x + 1.$$
This is even if $n$ is odd, so the "color difference" is $0$. In the second case for the minimum, we get
$$x + 3n - x + 1 = 3n + 1,$$
which is even if $n$ is odd, so the color difference is again $0$. In the third case we get
$$4n - x,$$
but the conditions $4n \leq 5n - x$ and $4n \leq x + 3n$ happen to imply $x = n$, so the color difference is actually $3n$, and this occurs in only one column. This is odd if $n$ is odd, so there is exactly one more plus sign than minus sign in this column (or the other way around).
This all means that the odd heart boards have a color difference of exactly two, one from column $x = n$, and one from column $x = -n$, so they cannot be tiled by dominoes. The even heart boards have an odd number of spaces, so they obviously cannot be tiled with dominoes.
