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)

Tutorial video below!

The Emperor needs you to prepare an auspicious floor for his tea ceremony, but the Ikea instructions only give the row and column projections of the tiling!  You must race the clock to reconstruct it before the Emperor has your head!

A tatami mat is a 1x1 or a 2x1 tile, used for traditional Japanese floorings.  In certain applications, it is "unlucky" to have four mats meeting at one point.

The game was conceived during a conversation with Martín Matamala , a mathematician from Universidad de Chile.  It is related to my PhD thesis work on the Tatami condition.  More details will be published in the upcoming journal version of "Auspicious tatami arrangements" by Frank Ruskey, Mark Schurch, Jennifer Woodock and myself.  A preprint should appear soon at http://arxiv.org/.

The in-game music is by Safary on ccmixter.org and the background photo is by 663highland on Wikipedia.

Here is an instructional video with wicked awesome Whaledub music from .  The music can be found at soundcloud.com - search for sness.

I appreciate your input and feedback in the comments.  Please link to the game through my site so I can measure the traffic before working on a commercial version.  If you are interested in doing the artwork for the commercial version, please contact me at alejandro.erickson@gmail.com

My tatami work at Moka House was briefly interrupted by a very enjoyable coffee with my friend Gwen Temmel today, during which she suggested that tatami tilings looked a little like the game Dots and Boxes.  A light went on inside me and I said,

LET’S PLAY DOTS AND BOXES WITH THE TATAMI CONDITION,

which just about as much inward enthusiasm.  The result was this:

If it is not obvious already, the game is played just as Dots and Boxes, but you may not create a four-valent site.   In other words, no pluses may be created.  If a player makes a box they must go again, no player may pass, the game ends when no legal move exists.

Playing this game is more fun, in my opinion than the original Dots and Boxes, and there is a mathematical curiosity that comes up.  Dots and Boxes ends when all bonds (edges) have been added, whereas this one has many different endings, each of them a maximal bond animal of a (typically) rectangular integer grid.  How many of these are there?

Notice that some tatami tilings are not maximal in this sense.  When a bidimer (formerly hamburger) has a long edge on the boundary of the grid, one of its dimers can be split into monomers.  I think this is the only such case, however.

Have you played Tatami Dots and Boxes?  Tell me about it in the comments!

What better way to start this tatami blog than with a bit of tatami art!  Here is a calligraphic illumination I created for the letter F.

This is an extension of Conway’s Game of Life in which there is more than one player (colour) and a fifth rule.  To win the game (or be winning) you must have more of your own cells on the board than your opponent does.

From Wikipedia, here are the rules to the original Game of Life

1. Any live cell with fewer than two live neighbours dies, as if caused by under-population.
2. Any live cell with more than three live neighbours dies, as if by overcrowding.
3. Any live cell with two or three live neighbours lives on to the next generation.
4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

Now we introduce colours and an extra rule.  Two two cells are friendly if they are the same colour:

1. Any live cell with fewer than two live friendly neighbours dies, as if by loneliness.
2. Any live cell with more than three live friendly neighbours dies, as if by overcrowding.
3. Any live cell with two or three friendly neighbours lives, unchanged, to the next generation (unless it dies at number 4).
4. Any live cell with strictly more enemy than friendly neighbours, dies.
5. Any dead cell with exactly three mutually friendly neighbours and no other live neighbours comes to life.

As you can see, when only the cells of one player are involved, these are the rules of the Game of Life.  When there are enemies, majority wins!