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ZOJ2507-Let's play a game(Anti-Nim)

Let‘s play a game

Time Limit: 2 Seconds      Memory Limit: 65536 KB

Elves from the Lothvain forest have created a very interesting game. The rules are very simple:

  • There are two players.

  • In the beginning there are n groups of stones. The i-th group contains a_i stones.

  • A player has to take a positive number of stones from exactly one of the groups in his turn.

  • A player who can‘t make a move (each group is empty) wins.

Very soon they have learnt how to always make the best possible move. Your task is to write a program that computes which of the players has a winning strategy for a given situation.

INPUT

The first number appearing in the input is number of datasets t (5<=t<=50). Each dataset is given in following format. The first row contains a single number n (1<=n<=10000). The next row contains exactly n numbers: a_1, a_2,..., a_n (0<=a_i<=1000000000). There are no empty lines between the datasets. You may assume that at least one group of stones is not empty.

OUTPUT

For each of datasets your program should write:

  • 1 if the first player has a winning strategy

  • 2 otherwise

in a separate row.

SAMPLE

INPUT
 
2
3
1 1 1
5
1 0 1 1 1
 
OUTPUT
2
1
题意:NIM游戏的变种,胜利状态变为不能取的取胜。
先手必胜当且仅当:
1、若所有堆的石子最大数量为1,且数量为1的堆数为偶数
2、若有的堆数量超过1,SG为非0
证明:情况一、很容易想到。
情况二、先分析只有一堆石子的数量大于1,那么这个人可以控制后手的1的数量,因此是必胜态(SG必为非0)。
其次,如果有大于1堆的石子的数量大于2,那么如果SG为非0,那么先手只要将SG变为0,后手无论进行什么操作,SG都会变成非0(异或的性质),这样进行下去,终会达到必胜态(只有一堆石子的最大数量大于1)。
#include <iostream>
#include <cstdio>
#include <cstring>
#include <vector>
#include <string>
#include <algorithm>
#include <queue>
using namespace std;
int main(){

    int ncase;
    cin >> ncase;
    while(ncase--){
        int n;
        int ans = 0;
        scanf("%d",&n);
        int cnt = 0;
        for(int i = 0; i < n; i++){
            int t;
            scanf("%d",&t);
            if(t>1) cnt++;
            ans ^= t;
        }
        if(cnt==0){
           if(ans){
                printf("2\n");
           }else{
                printf("1\n");
           }
        }
        else{
            if(ans) printf("1\n");
            else printf("2\n");
        }

    }
    return 0;
}