Dots and Boxes

A grid of dots. Draw a line between two of them that are next to each other; if the line completes a box, the box is yours and you draw again. Twenty-five boxes on this board, so there are no draws — somebody ends up with thirteen.

It is the schoolbook game everybody played in the back of an exercise book, and it has a piece of real mathematics hiding in it. Once the board fills up, every remaining line either is safe or hands the other player a run of boxes. The safe moves run out. Whoever runs out of them first has to open a chain, and whoever opens a chain gives away everything in it. So the game is not really about taking boxes at all. It is about who is left holding the board when the safe moves are gone.

Three opponents. Easy takes every box it can see and then draws at random. Medium knows what a safe line is and opens the shortest chain when it has to. Hard adds the move that beats both of them — it will take all but two boxes of a chain and hand the rest back, on purpose, to make you open the next one — and once a dozen lines are left it stops guessing and plays the position out exactly.