（SG） poj 2960

Arthur and his sister Caroll have been playing a game called Nim for some time now. Nim is played as follows:

• The starting position has a number of heaps, all containing some, not necessarily equal, number of beads.
• The players take turns chosing a heap and removing a positive number of beads from it.
• The first player not able to make a move, loses.

Arthur and Caroll really enjoyed playing this simple game until they 
recently learned an easy way to always be able to find the best move:

• Xor the number of beads in the heaps in the current position (i.e. if we have 2, 4 and 7 the xor-sum will be 1 as 2 xor 4 xor 7 = 1).
• If the xor-sum is 0, too bad, you will lose.
• Otherwise, move such that the xor-sum becomes 0. This is always possible.

It is quite easy to convince oneself that this works. Consider these facts:

• The player that takes the last bead wins.
• After the winning player‘s last move the xor-sum will be 0.
• The xor-sum will change after every move.

Which means that if you make sure that the xor-sum always is 0 when you have made your move, your opponent will never be able to win, and, thus, you will win. 

Understandibly it is no fun to play a game when both players know how to play perfectly (ignorance is bliss). Fourtunately, Arthur and Caroll soon came up with a similar game, S-Nim, that seemed to solve this problem. Each player is now only allowed to remove a number of beads in some predefined set S, e.g. if we have S = {2, 5} each player is only allowed to remove 2 or 5 beads. Now it is not always possible to make the xor-sum 0 and, thus, the strategy above is useless. Or is it? 

your job is to write a program that determines if a position of S-Nim is a losing or a winning position. A position is a winning position if there is at least one move to a losing position. A position is a losing position if there are no moves to a losing position. This means, as expected, that a position with no legal moves is a losing position.

Input consists of a number of test cases. 
For each test case: The first line contains a number k (0 < k ≤ 100) describing the size of S, followed by k numbers si (0 < si ≤ 10000) describing S. The second line contains a number m (0 < m ≤ 100) describing the number of positions to evaluate. The next m lines each contain a number l (0 < l ≤ 100) describing the number of heaps and l numbers hi (0 ≤ hi ≤ 10000) describing the number of beads in the heaps. 
The last test case is followed by a 0 on a line of its own.

For each position: If the described position is a winning position print a ‘W‘.If the described position is a losing position print an ‘L‘. 
Print a newline after each test case.

2 2 5
3
2 5 12
3 2 4 7
4 2 3 7 12
5 1 2 3 4 5
3
2 5 12
3 2 4 7
4 2 3 7 12
0

LWW
WWL

#include<iostream>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<cstdlib>
#include<string>
#include<algorithm>
using namespace std;
int num[105],k,m,n,res;
int sg[10010];
int SG(int x)
{
    int i;
    if(sg[x]+1) return sg[x];
    bool vis[10010]={0};
    for(i=1;i<=k;i++)
    {
        if(x>=num[i])
            vis[SG(x-num[i])]=1;
    }
    for(i=0;vis[i];i++);
    return sg[x]=i;

}
int main()
{
    while(scanf("%d",&k)!=EOF)
    {
        if(k==0)
            break;
        for(int i=1;i<=k;i++)
            scanf("%d",&num[i]);
        memset(sg,-1,sizeof(sg));
        sg[0]=0;
        scanf("%d",&m);
        while(m--)
        {
            int nim=0;
            scanf("%d",&n);
            for(int i=1;i<=n;i++)
            {
                scanf("%d",&res);
                nim=nim^SG(res);
            }
            if(nim)
                printf("W");
            else
                printf("L");
        }
        printf("\n");
    }
    return 0;
}