Counting to 21 Strategy

21 is a simple game that you can always win if you go second. The goal of the game is to be the first person to say “21”. The rules are that you can only add 1 or 2 to whatever the other player says.

For example, the first player can say “1” or “2”. If he said “2” the second player can say “3” or “4” and so on.

There is a strictly dominant strategy in this game. You can use backward induction to figure out what to do. Try to figure it out yourself first, but below is the strategy.

  1. You need to get them to say 19 or 20; therefore, if you say 18, you win.
  2. To say 18, you need to get them to say 16, or 17, so if you say 15, you win.
  3. Continuing on, if you say 12 you win
  4. if you say 9, you win,
  5. if you say 6, you win,
  6. if you say 3, you win
  7. So, if you go second, you can guarantee that you will say 3, and win every time.

When playing this with people, the secret is to not let on that you are doing a strategy against them. To win the most times possible, act like you are thinking about what to do on every turn, even though you know what you will say.

If you go first, just hope that they don’t know the dominant strategy and mess up one time. Once that happens, you can get back onto the winning track and win every time.

There are different variations where you lose if you say 21. Can you figure out the dominant strategy in that game?



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