In a variant of the game of Nim, two players alternate taking turns in removing 1, 2, or 3 beans from a pile. The object is to remove the last bean. If the initial pile has 29 beanss, should you want to go first or second to guarantee a win?
The player that goes 1st gets all of
the odd numbers including # 29.
To determine whether you should go first or second to guarantee a win in this variant of the game, we can analyze the situation.
In this game, the goal is to force your opponent into a losing position, where they have to take the last bean. We can start by observing the pattern that arises when we consider the number of beans remaining after each turn:
If the number of beans remaining after a player's turn is a multiple of 4 (ex. 0, 4, 8, 12, etc.), then that player will always lose if both players play optimally.
If the number of beans remaining after a player's turn is not a multiple of 4, then that player can force a win if they play optimally.
So in this case, since the initial pile has 29 beans, which is not a multiple of 4, it means the first player can force a win if they play optimally.
Hence, to guarantee a win, you should want to go first in this variant of the game.