My PRIMES STEP program consists of two groups of ten students each: the senior group and the junior group. The senior group is usually stronger, and they were especially productive last academic year. We wrote four papers, which I described in the post EvenQuads at PRIMES STEP. The junior group wrote one paper related to the game SOS. The game was introduced in the following 1999 USAMO problem.

Problem. The game is played on a 1-by-2000 grid. Two players take turns writing an S or an O in an empty square. The first player who produces three consecutive squares that spell SOS wins. The game is a draw if all squares are filled without producing SOS. Prove that the second player has a winning strategy.

The solution is quite pretty, so I do not want to spoil it. If my readers want it, the solution for this grid, and, more generally, for any grid of size 1-by-n, is posted in many places.

My students studied generalizations of this game, and the results are posted at the arXiv: SOS. We tried different target strings and showed that:

  • The SOO game is always a draw.
  • The SSS game is always a draw.
  • The SOSO game is always a draw.

Then, we tried a version where the winner needed to spell one of two target strings. We showed that:

  • The SSSS-OOOO game is always a draw.

We tried several more elaborate variations, but I want to keep this post short.


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