I think a significant start to my development as a mathematician was playing card games (mostly Euchre) with my parents in my youth. After a particular round, my father would tell me, "Well, with your hand, you should have done this. It would have forced me to do that. And then you could've come back with this, thus winning the round." Basically, I feel like I was taught early on how to reason and make an argument in the context of a game.
Fast forward, next semester I'm teaching an "intro to proofs" course (not for the first time) and I'd really like to start the semester off with having the students play a game in pairs such that they can explain after each round who should have won. Unfortunately, most of the games I know and enjoy playing are too complicated or too long to achieve this efficiently.
So my question is: Does anyone know of a game that satisfies some or all of
- Easy to learn (so it doesn't take all class to explain it).
- Rounds are short (so multiple rounds can be played and students can get a sense of how it works).
- Easy to move back and forth in time (so it's easy to replay the game from various points, so probably only a handful of moves in each round).
- Sometimes, if the opponent plays a certain way, your choice of moves may be reduced to only one option.
- Conducive to basic logical analysis.
The only idea I've had so far is Tic-Tac-Toe, but I thought I'd ask to see if there's maybe a lesser-known game that also fits the bill.