组合数学取子游戏nim
4星 · 超过85%的资源 需积分: 15 158 浏览量
2013-03-31
14:32:57
上传
评论
收藏 19KB DOC 举报 
nim
Time Limit:1000MS Memory Limit:65536K
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 removing one 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, …, kn stones respectively; in such a position, there are k1 + k2 +
… + kn possible moves. We write each ki in binary (base 2). Then, the Nim position is losing if
and only if, among all the ki’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 the xor of the ki’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, suppose k3 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 when k1 = 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 ≤ ki ≤ 1, 000,
000, 000, indicating the number of stones in each pile. The end-of-file is marked by a test case
with n = 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.
资源评论
a18942337162013-10-06这个代码真的有点少，有点简单，不过是能用
