Hex

Nash proved in 1949 that Hex can't end in a draw - a non-trivial result colloquially called the "Hex theorem", which we now know is equivalent to the Brouwer fixed-point theorem.

Nash also proved that the first player in Hex on a board of any size can force a win. However, the actual winning strategy for 11×11 Hex is still unknown.

In 11×11 Hex, there are approximately 2.4e56 legal positions - about 5 billion times more than in chess!
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How to play:

1. Players take turns flipping the hexagonal cells to become their colour.

2. The first player to form a connected path linking the opposing sides of the board marked by their colour wins.

3. The four corner hexagons belong to both adjacent sides.

4. Swap rule: after the first move, the second player may decide to play as the first player from then on.

5. If you can think of a strong response to your own move then look for a better one!