310484: CF1841A. Game with Board
Description
Alice and Bob play a game. They have a blackboard; initially, there are $n$ integers written on it, and each integer is equal to $1$.
Alice and Bob take turns; Alice goes first. On their turn, the player has to choose several (at least two) equal integers on the board, wipe them and write a new integer which is equal to their sum.
For example, if the board currently contains integers $\{1, 1, 2, 2, 2, 3\}$, then the following moves are possible:
- choose two integers equal to $1$, wipe them and write an integer $2$, then the board becomes $\{2, 2, 2, 2, 3\}$;
- choose two integers equal to $2$, wipe them and write an integer $4$, then the board becomes $\{1, 1, 2, 3, 4\}$;
- choose three integers equal to $2$, wipe them and write an integer $6$, then the board becomes $\{1, 1, 3, 6\}$.
If a player cannot make a move (all integers on the board are different), that player wins the game.
Determine who wins if both players play optimally.
InputThe first line contains one integer $t$ ($1 \le t \le 99$) — the number of test cases.
Each test case consists of one line containing one integer $n$ ($2 \le n \le 100$) — the number of integers equal to $1$ on the board.
OutputFor each test case, print Alice if Alice wins when both players play optimally. Otherwise, print Bob.
ExampleInput2 3 6Output
Bob AliceNote
In the first test case, $n = 3$, so the board initially contains integers $\{1, 1, 1\}$. We can show that Bob can always win as follows: there are two possible first moves for Alice.
- if Alice chooses two integers equal to $1$, wipes them and writes $2$, the board becomes $\{1, 2\}$. Bob cannot make a move, so he wins;
- if Alice chooses three integers equal to $1$, wipes them and writes $3$, the board becomes $\{3\}$. Bob cannot make a move, so he wins.
In the second test case, $n = 6$, so the board initially contains integers $\{1, 1, 1, 1, 1, 1\}$. Alice can win by, for example, choosing two integers equal to $1$, wiping them and writing $2$ on the first turn. Then the board becomes $\{1, 1, 1, 1, 2\}$, and there are three possible responses for Bob:
- if Bob chooses four integers equal to $1$, wipes them and writes $4$, the board becomes $\{2,4\}$. Alice cannot make a move, so she wins;
- if Bob chooses three integers equal to $1$, wipes them and writes $3$, the board becomes $\{1,2,3\}$. Alice cannot make a move, so she wins;
- if Bob chooses two integers equal to $1$, wipes them and writes $2$, the board becomes $\{1, 1, 2, 2\}$. Alice can continue by, for example, choosing two integers equal to $2$, wiping them and writing $4$. Then the board becomes $\{1,1,4\}$. The only possible response for Bob is to choose two integers equal to $1$ and write $2$ instead of them; then the board becomes $\{2,4\}$, Alice cannot make a move, so she wins.
Input
题意翻译
Alice 和 Bob 玩游戏，他们有一块黑板。最初，有 $n$ 个整数 $1$。Alice 和 Bob 轮流操作，Alice 先手。 轮到时，玩家必须在棋盘上选择几个（至少两个）相等的整数，擦除它们，然后写一个等于它们总和的新整数。 如果玩家不能移动（棋盘上的所有整数都不同），该玩家将赢得游戏。 如果两名选手都发挥最佳，则确定谁获胜。Output
Alice和Bob在黑板上玩一个游戏，最初黑板上写有n个整数，每个整数都是1。Alice和Bob轮流进行操作，Alice先手。在每一轮中，玩家必须选择至少两个相同的整数，擦除它们，并在黑板上写一个新的整数，这个整数等于它们之和。例如，如果黑板上当前包含整数{1, 1, 2, 2, 2, 3}，则可能的移动有：选择两个整数1，擦除它们并写一个整数2，然后黑板变成{2, 2, 2, 2, 3}；选择两个整数2，擦除它们并写一个整数4，然后黑板变成{1, 1, 2, 3, 4}；选择三个整数2，擦除它们并写一个整数6，然后黑板变成{1, 1, 3, 6}。如果一个玩家无法进行移动（黑板上的所有整数都不同），那么那个玩家赢得游戏。确定如果两个玩家都进行最优游戏，谁会赢。
输入输出数据格式：
输入：
第一行包含一个整数t（1≤t≤99），表示测试用例的数量。
每个测试用例由一行组成，包含一个整数n（2≤n≤100），表示黑板上等于1的整数的数量。
输出：
对于每个测试用例，如果Alice在两个玩家都进行最优游戏时获胜，则打印“Alice”。否则，打印“Bob”。题目大意： Alice和Bob在黑板上玩一个游戏，最初黑板上写有n个整数，每个整数都是1。Alice和Bob轮流进行操作，Alice先手。在每一轮中，玩家必须选择至少两个相同的整数，擦除它们，并在黑板上写一个新的整数，这个整数等于它们之和。例如，如果黑板上当前包含整数{1, 1, 2, 2, 2, 3}，则可能的移动有：选择两个整数1，擦除它们并写一个整数2，然后黑板变成{2, 2, 2, 2, 3}；选择两个整数2，擦除它们并写一个整数4，然后黑板变成{1, 1, 2, 3, 4}；选择三个整数2，擦除它们并写一个整数6，然后黑板变成{1, 1, 3, 6}。如果一个玩家无法进行移动（黑板上的所有整数都不同），那么那个玩家赢得游戏。确定如果两个玩家都进行最优游戏，谁会赢。 输入输出数据格式： 输入： 第一行包含一个整数t（1≤t≤99），表示测试用例的数量。 每个测试用例由一行组成，包含一个整数n（2≤n≤100），表示黑板上等于1的整数的数量。 输出： 对于每个测试用例，如果Alice在两个玩家都进行最优游戏时获胜，则打印“Alice”。否则，打印“Bob”。