# 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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