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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