codeforces_346A Alice and Bob(数学)

Posted 2020-09-17 EricJeffrey

题目链接:http://codeforces.com/problemset/problem/346/A

参考链接:http://blog.csdn.net/loy_184548/article/details/50174615

感受到数学在博弈论中的强大。

考虑最后终止状态的序列-无法取出任意两个数他们的差值不存在这个序列中:那么这必定是个首项等于公差的等差序列

而这个序列是 d 2d 3d....，因此可以通过a[1] a[2] a[3] ...的最大公约数得到

然后计算有几个数没在数组中，判断奇偶性即可

代码:

 1 #include <cstdio>
 2 #include <cstdlib>
 3 #include <iostream>
 4 #include <cstring>
 5 #include <cmath>
 6 #include <algorithm>
 7 #include <set> 
 8 using namespace std;
 9 int a[105];
10 
11 int gcd(int x, int y){
12     return x == 0? y : gcd(y % x, x);
13 }
14 
15 int main(){
16     int n, maxn = 0;
17     scanf("%d", &n);
18     for(int i = 0; i < n; i++){
19         scanf("%d", &a[i]);
20         maxn = max(a[i], maxn);
21     }
22     int cnt = 0;
23     int g = a[0];
24     for(int i = 1; i < n; i++){
25         g = gcd(a[i], g);
26     }
27     cnt = maxn / g;
28     for(int i = 0; i < n; i++){
29         if(a[i] % g == 0) cnt--;
30     }
31     if(cnt & 1){
32         printf("Alice\n");
33     }
34     else{
35         printf("Bob\n");
36     }
37 }

题目：

A. Alice and Bob

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

It is so boring in the summer holiday, isn‘t it? So Alice and Bob have invented a new game to play. The rules are as follows. First, they get a set ofn distinct integers. And then they take turns to make the following moves. During each move, either Alice or Bob (the player whose turn is the current) can choose two distinct integers x and y from the set, such that the set doesn‘t contain their absolute difference |x?-?y|. Then this player adds integer |x?-?y| to the set (so, the size of the set increases by one).

If the current player has no valid move, he (or she) loses the game. The question is who will finally win the game if both players play optimally. Remember that Alice always moves first.

Input

The first line contains an integer n (2?≤?n?≤?100) — the initial number of elements in the set. The second line contains n distinct space-separated integers a1,?a2,?...,?an (1?≤?ai?≤?109) — the elements of the set.

Output

Print a single line with the winner‘s name. If Alice wins print "Alice", otherwise print "Bob" (without quotes).

Examples

input

2
2 3

output

Alice

input

2
5 3

output

Alice

input

3
5 6 7

output

Bob

Note

Consider the first test sample. Alice moves first, and the only move she can do is to choose 2 and 3, then to add 1 to the set. Next Bob moves, there is no valid move anymore, so the winner is Alice.