I had a word with Ron Graham at the 2010 CMS Winter Meeting and he mentioned a short paper of his in which he talks about fault-free dimer tilings. A fault-free tiling of a rectangular grid is one in which every interior grid line intersects at least one tile. Here is a 6 x 6 tiling with two faults.
Ron Graham gives a clever argument, due to S. W. Golomb and R. I. Jewett, that no 6 x 6 fault-free dimer tiling exists. Try to prove it yourself! (hint: a dimer blocks exactly one potential fault. Use elementary arguments).
Let's ask the obvious question.
For which dimensions r x c can we have fault free tatami-dimer tilings?
The answer is simple. It never happens, except, in the 1 x 2 rectangle. I'll allow you to convince yourself (in keeping with the low production value of this post) by looking at this:
Dimer tatami tilings of rectangles are all compositions of these configurations. Since their compositions create fault lines, we may assume that a fault-free tiling has a single bidimer (the configuration in the middle). But each of these clearly has at least one fault line too.
(disclaimer: I'm posting this before I'm done thinking about it. If you want to cite it you should contact me and I'll write a better proof)