In a variant of the game 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 beans, should you want to go first or second to guarantee a win?
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To determine whether you should want to go first or second in this variant of the game Nim, you can analyze the situation using a mathematical strategy. The key idea is understanding the concept of "winning" and "losing" positions in the game.
First, let's observe some patterns:
- If the number of beans in the pile is a multiple of 4, the starting player (Player 1) will always lose if both players play optimally.
- If the number of beans in the pile is not a multiple of 4, the starting player (Player 1) can always win if they play optimally.
In this case, the initial pile has 29 beans, which is not a multiple of 4. Therefore, if both players play optimally, the starting player (Player 1) can guarantee a win.
To understand why this is true, you can use a backward reasoning approach:
- If there are 29 beans, Player 1 can remove 1 bean, leaving 28 beans in the pile.
- Now, Player 2 can only remove 1, 2, or 3 beans. Regardless of the number of beans Player 2 removes, Player 1 can always remove the remaining beans (leaving a multiple of 4 beans) and force Player 2 into a losing position.
- By continuing this backward reasoning, you'll find that Player 1 can always mirror Player 2's moves, eventually guaranteeing a win.
Therefore, in this variant of Nim with 29 beans, you should want to go first to guarantee a win if both players play optimally.