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原标题:LeetCode -- Nim Game

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LeetCode -- Nim Game

题目描述:

You are playing the following Nim Game with your friend: There is a heap of stones on the table, each time one of you take turns to remove 1 to 3 stones. The one who removes the last stone will be the winner. You will take the first turn to remove the stones.

Both of you are very clever and have optimal strategies for the game. Write a function to determine whether you can win the game given the number of stones in the heap.

For example, if there are 4 stones in the heap, then you will never win the game: no matter 1, 2, or 3 stones you remove, the last stone will always be removed by your friend.

一个拿石子的游戏,有n个石子,每次可以从中拿走[1,3]个。你和另一个人轮流拿石子,你先拿。
问:对于数量为n的石子,能否判断你必胜?

思路:
如果有4个石子,轮到谁谁就输了;
有8个石子,轮到谁谁也输,因为我无论拿几个,对方都会有办法给我剩4个;
以此类推,有4k个石子,轮到谁谁输。

而由于我先拿,因此只要第一次我能保证给对方剩余4的倍数个石子,我就能赢。
这样就等于把这个拿石子问题转化为了判断n是否为4的倍数的问题。

因此代码很简单:

 public bool CanWinNim(int n) {
        if(n < 4){
      return true;
     }

     return n % 4 != 0;
    }

 

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[LeetCode从零单刷]Nim Game

题目:

 

You are playing the following Nim Game with your friend: There is a heap of stones on the table, each time one of you take turns to remove 1 to 3 stones. The one who removes the last stone will be the winner. You will take the first turn to remove the stones.

Both of you are very clever and have optimal strategies for the game. Write a function to determine whether you can win the game given the number of stones in the heap.

For example, if there are 4 stones in the heap, then you will never win the game: no matter 1, 2, or 3 stones you remove, the last stone will always be removed by your friend.

解答:

博弈论中极为经典的尼姆游戏。有总数为n的石头,每个人可以拿1~m个石头,两个人交替拿,拿到最后一个的人获胜。究竟是先手有利,还是后手有利?

 

1个石子,先手全部拿走;2个石子,先手全部拿走;3个石子,先手全部拿走;4个石子,后手面对的是先手的第1,2,3情况,后手必胜;5个石子,先手拿走1个让后手面对第4种情况,后手必败;6个石子,先手拿走2个让后手面对第4种情况,后手必败;…… 容易看出来,只有当出现了4的倍数,先手无可奈何,其余情况先手都可以获胜。 (石子数量为4的倍数)后手的获胜策略十分简单,每次取石子的数量,与上一次先手取石子的数量和为4即可; (石子数量不为4的倍数)先手的获胜策略也十分简单,每次都令取之后剩余的石子数量为4的倍数(4*0=0,直接拿光),他就处于后手的位置上,利用上一行的策略获胜。
代码极为简单:

class Solution {
public:
    bool canWinNim(int n) {
        if(n % 4 == 0)  return false;
        else    return true;
    }
};

 

 

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