Hex cannot end in a draw

Theorem: Hex cannot end in a draw.

There are several ways of proving this. There is a proof by induction, but it is rather messy. The two that I like the best are:

  • A proof by David Gale that exploits the fact that exactly three hexes meet at every vertex. This property is very fruitful; hex tilings really are much "better" than any other tilings!
  • A very elegant proof using the game of Y, a more general game than Hex (see the related games page) that shares many of its properties.

For the savvy reader: The fact that a completely filled Hex board must contain a winning chain is equivalent to Brouwer's fixed point theorem, that says that any continuous mapping from the unit square onto itself must contain a point that gets mapped to itself. This is a substantially nontrivial theorem in topology.