Alice and Bob Playing Flower Game

Problem Description

Alice and Bob are playing a turn-based game on a circular field surrounded by flowers. The game consists of two players choosing flowers from either side of a circular arrangement, and the player who picks the last flower wins. Given the number of flowers in the clockwise direction (x) and anti-clockwise direction (y), the task is to compute the number of possible pairs (x, y) such that Alice has a winning strategy.

Key Insights

• Alice goes first and has a choice of picking from either direction.
• The game ends when there are no flowers left, and the player who made the last pick wins.
• Alice must ensure that she can always force a win regardless of Bob's moves.
• The conditions for Alice to win can be derived from the relative sizes of x and y.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we need to analyze the conditions under which Alice can guarantee a win. The key observation is:

• If Alice can make sure that she can always leave Bob with a situation where he cannot win, it usually involves the counts of flowers being unequal or having certain strategic advantages based on their counts.

The algorithm can be outlined as follows:

1. Iterate through all possible values of x from 1 to n.
2. For each x, calculate the minimum value of y such that Alice can win.
3. Count valid pairs (x, y) where y is within the allowed range [1, m].

We can use a mathematical approach to derive how many pairs (x, y) satisfy the winning condition for Alice.

Code Solutions