sg函数的变形

Nim is a two-player mathematic game of strategy in which players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap.

Nim is usually played as a misere game, in which the player to take the last object loses. Nim can also be played as a normal play game, which means that the person who makes the last move (i.e., who takes the last object) wins. This is called normal play because most games follow this convention, even though Nim usually does not.

Alice and Bob is tired of playing Nim under the standard rule, so they make a difference by also allowing the player to separate one of the heaps into two smaller ones. That is, each turn the player may either remove any number of objects from a heap or separate a heap into two smaller ones, and the one who takes the last object wins.

InputInput contains multiple test cases. The first line is an integer 1 ≤ T ≤ 100, the number of test cases. Each case begins with an integer N, indicating the number of the heaps, the next line contains N integers s[0], s[1], ...., s[N-1], representing heaps with s[0], s[1], ..., s[N-1] objects respectively.(1 ≤ N ≤ 10^6, 1 ≤ S[i] ≤ 2^31 - 1)OutputFor each test case, output a line which contains either "Alice" or "Bob", which is the winner of this game. Alice will play first. You may asume they never make mistakes.Sample Input

2
3
2 2 3
2
3 3

Sample Output

Alice
Bob

题目分析：正常的nim游戏规则，但是操作的过程中可以不取出石子，但是可以将石子一分为二。
思路分析：打表找一下所有数的sg，发现是有规律的
代码示例：

打表：

void init(){
    sg[0] = 0, sg[1] = 1;
    
    for(int i = 2; i <= 100; i++){
        memset(s, 0, sizeof(s));
        for(int j = 0; j < i; j++){
            s[sg[j]] = 1;
        }    
        for(int j = 1; j <= (i/2); j++){
            s[sg[j]^sg[i-j]] = 1;
        }
        for(int j = 0; ;j++){
            if(!s[j]) {sg[i] = j; break;}
        }
    }
}

 AC代码：

int x;
int n;

int sg(int x){
    int k = x%4;
    if (k == 1 || k == 2) return x;
    else if (k == 3) return x+1;
    else return x-1;
}

int main() {
    //freopen("in.txt", "r", stdin);
    //freopen("out.txt", "w", stdout);
    int t;
    
    cin >>t;
    while(t--){
        scanf("%d", &n);
        int s = 0;
        for(int i = 1; i <= n; i++){
            scanf("%d", &x); 
            s ^= sg(x);
        }        
        
        if (s) printf("Alice
");
        else printf("Bob
"); 
    }
    return 0;
}