[poj2975]Nim

最新推荐文章于 2021-09-05 15:42:50 发布

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最新推荐文章于 2021-09-05 15:42:50 发布

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文章标签： each pascal integer input output

本文链接：https://blog.youkuaiyun.com/hccz95/article/details/6286327

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Nim

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 2835		Accepted: 1232

Description

Nim is a 2-player game featuring several piles of stones. Players alternate turns, and on his/her turn, a player’s move consists of removingone or more stones from any single pile. Play ends when all the stones have been removed, at which point the last player to have moved is declared the winner. Given a position in Nim, your task is to determine how many winning moves there are in that position.

A position in Nim is called “losing” if the first player to move from that position would lose if both sides played perfectly. A “winning move,” then, is a move that leaves the game in a losing position. There is a famous theorem that classifies all losing positions. Suppose a Nim position contains n piles having k₁,k₂, …,k_n stones respectively; in such a position, there arek₁ +k₂ + … + k_n possible moves. We write eachk_i in binary (base 2). Then, the Nim position is losing if and only if, among all thek_i’s, there are an even number of 1’s in each digit position. In other words, the Nim position is losing if and only if thexor of thek_i’s is 0.

Consider the position with three piles given by k₁ = 7, k₂ = 11, and k₃ = 13. In binary, these values are as follows:

 111
1011
1101

There are an odd number of 1’s among the rightmost digits, so this position is not losing. However, supposek₃ were changed to be 12. Then, there would be exactly two 1’s in each digit position, and thus, the Nim position would become losing. Since a winning move is any move that leaves the game in a losing position, it follows that removing one stone from the third pile is a winning move whenk₁ = 7, k₂ = 11, and k₃ = 13. In fact, there are exactly three winning moves from this position: namely removing one stone from any of the three piles.

Input

The input test file will contain multiple test cases, each of which begins with a line indicating the number of piles, 1 ≤n ≤ 1000. On the next line, there are n positive integers, 1 ≤k_i ≤ 1, 000, 000, 000, indicating the number of stones in each pile. The end-of-file is marked by a test case withn = 0 and should not be processed.

Output

For each test case, write a single line with an integer indicating the number of winning moves from the given Nim position.

Sample Input

3
7 11 13
2
1000000000 1000000000
0

Sample Output

3
0

Source

Stanford Local 2005

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题意:

题意跟这道题一样: http://blog.youkuaiyun.com/hccz95/archive/2011/03/29/6286266.aspx

只是求的不是简单的是否有必胜策略,而是要求先手第一步有几种情况是必胜策略.

只要求出 sum=a[1] xor a[2] xor...xor a[n],

若sum=0,则先手必输;

若sum>0,先手可以进行操作是sum = 0,到达一个先手必输态,操作如下:

设 a[i]' = sum xor a[i], 若a[i]' < a[i], 那么可以把a[i]变成a[i], 此时 sum'=sum xor a[i] xor a[i]'=0