# Twenty One or Two Hundred Thirty Nine?

Logic Level 2

Alice and Carla are playing a game often learned in elementary school known as Twenty One. The rules for the game are as follows:

• Each player takes turns saying between $$1$$ and $$3$$ consecutive numbers, with the first player starting with the number $$1$$. For example, Player $$1$$ could say the numbers $$1$$ and $$2$$, then Player $$2$$ can say "$$3$$, $$4$$, $$5$$", then Player $$1$$ can say "$$6$$" and so on.

• The goal of the game is to get the other person to say "$$21$$", meaning that you have to be the one to say "$$20$$".

Carla begins to get bored with the game, so she decides to make it a little bit more challenging. The name of the new game is Two Hundred Thirty Nine, where the goal is now to get the other person to say "$$239$$". Carla decides that she'll go first and that Alice will go second. Also, each player is now able to say up to "$$6$$" numbers per turn. Is there a way to tell which player is going to win before the game even starts?

Details and Assumptions:

• Assume that each player plays "perfectly", meaning that if there was an optimal way of playing, both players would be playing the best that the game allows them to play.
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