Statistical mechanics of tetris

I'm finding that I'm becoming increasingly fascinated by shape. It seems such a simple thing yet scratch the surface only a little and the complexity comes pouring out. Take simple tiling problems; I can tile my floor with squares or regular hexagons, but not regular octagons - they'll always leave annoying gaps. From a statistical mechanics point of view those gaps are very important, little sources of entropy that you can't get rid of. In three dimensions understanding the packing of tetrahedra has proved no simple task. But that's a story for another day.

So it came as no surprise that I was very taken with Lev Gelb's talk on polyominoes at the Brno conference. Polyominoes are connected shapes on a two dimensional lattice. A monomino is a square, a domino you know. Tetrominoes are made of four squares and are exactly like the pieces from Tetris. Assuming that they're stuck in the plane (so you can't flip them over) there are 7 tetrominoes.

Tree diagrams solve everything

Just a quick one. I saw this post, When intuition and math probably look wrong, via Ben Goldacre's mini blog. The problem is set as follows:

I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?

Intuition tells you the answer is 1/2, mathematicians tell you it's something else. I'll leave the answer until the end of the post in case you want to run off and solve it first. It's essentially a fancier version of the Monty Hall problem.