CF1841A.Game with Board

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.

The 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.

For each test case, print Alice if Alice wins when both players play optimally. Otherwise, print Bob.

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.