Eat the Chip
Contest: Codeforces Round 920 (Div. 3)
Difficulty rating: 1600
Tags: brute force, games, greedy, math

Alice and Bob are playing a game on a checkered board. The board has $$$h$$$ rows, numbered from top to bottom, and $$$w$$$ columns, numbered from left to right. Both players have a chip each. Initially, Alice's chip is located at the cell with coordinates $$$(x_a, y_a)$$$ (row $$$x_a$$$, column $$$y_a$$$), and Bob's chip is located at $$$(x_b, y_b)$$$. It is guaranteed that the initial positions of the chips do not coincide. Players take turns making moves, with Alice starting.

On her turn, Alice can move her chip one cell down or one cell down-right or down-left (diagonally). Bob, on the other hand, moves his chip one cell up, up-right, or up-left. It is not allowed to make moves that go beyond the board boundaries.

More formally, if at the beginning of Alice's turn she is in the cell with coordinates $$$(x_a, y_a)$$$, then she can move her chip to one of the cells $$$(x_a + 1, y_a)$$$, $$$(x_a + 1, y_a - 1)$$$, or $$$(x_a + 1, y_a + 1)$$$. Bob, on his turn, from the cell $$$(x_b, y_b)$$$ can move to $$$(x_b - 1, y_b)$$$, $$$(x_b - 1, y_b - 1)$$$, or $$$(x_b - 1, y_b + 1)$$$. The new chip coordinates $$$(x', y')$$$ must satisfy the conditions $$$1 \le x' \le h$$$ and $$$1 \le y' \le w$$$.

Example game state. Alice plays with the white chip, Bob with the black one. Arrows indicate possible moves.

A player immediately wins if they place their chip in a cell occupied by the other player's chip. If either player cannot make a move (Alice—if she is in the last row, i.e. $$$x_a = h$$$, Bob—if he is in the first row, i.e. $$$x_b = 1$$$), the game immediately ends in a draw.

What will be the outcome of the game if both opponents play optimally?

Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. This is followed by the description of the test cases.

Each test case consists of a single line containing six integers $$$h$$$, $$$w$$$, $$$x_a$$$, $$$y_a$$$, $$$x_b$$$, $$$y_b$$$ ($$$1 \le x_a, x_b \le h \le 10^6$$$, $$$1 \le y_a, y_b \le w \le 10^9$$$) — the dimensions of the board and the initial positions of Alice's and Bob's chips. It is guaranteed that either $$$x_a \ne x_b$$$ or $$$y_a \ne y_b$$$.

It is guaranteed that the sum of $$$h$$$ over all test cases does not exceed $$$10^6$$$.

Output format:
For each test case, output "Alice" if Alice wins, "Bob" if Bob wins, and "Draw" if neither player can secure a victory. You can output each letter in any case (lowercase or uppercase). For example, the strings "bOb", "bob", "Bob", and "BOB" will be accepted as Bob's victory.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Example input:
12
6 5 2 2 5 3
4 1 2 1 4 1
1 4 1 3 1 1
5 5 1 4 5 2
4 4 1 1 4 4
10 10 1 6 10 8
10 10 2 6 10 7
10 10 9 1 8 1
10 10 8 1 10 2
10 10 1 1 2 1
10 10 1 3 4 1
10 10 3 1 1 1

Example output:
Alice
Bob
Draw
Draw
Draw
Alice
Draw
Draw
Bob
Alice
Alice
Draw

Editorial:
First, let's note that the difference $$$x_b - x_a$$$ decreases exactly by one each (both Alice's and Bob's) turn. In the end, if one of the players was able to win, $$$x_b - x_a = 0$$$. In particular, that means that if $$$x_a - x_b$$$ is initially odd then only Alice has a chance of winning the match and vice versa.
Suppose that $$$x_a - x_b$$$ is initially even (the outcome of the match could be either Bob's win or draw). If $$$x_a \ge x_b$$$ the answer is immediately draw. Otherwise, the players will make $$$t = (x_b - x_a) / 2$$$ moves each before $$$x_a = x_b$$$.
If at some point during these $$$2t$$$ moves Bob can achieve $$$y_a = y_b$$$, he is winning as he can continue with symmetrical responses to Alice's turns.
If $$$y_a > y_b$$$ and Bob cannot reach right border ($$$w > y_b + t$$$), Alice can always choose the rightmost option for her and after each of $$$2t$$$ moves $$$y_a$$$ will be greater than $$$y_b$$$ which means Bob cannot win. Otherwise, if Bob always chooses the rightmost option for him, he will eventually achieve $$$y_a = y_b$$$.
The case when $$$y_a$$$ is initially less than $$$y_b$$$ as well as the case when Alice has a chance to win ($$$x_b - x_a$$$ is odd) can be covered in a similar way.