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