Introduction

In this blog post, we will explore how to solve the LeetCode problem "2550" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

There is a regular convex polygon with n vertices. The vertices are labeled from 0 to n - 1 in a clockwise direction, and each vertex has exactly one monkey. The following figure shows a convex polygon of 6 vertices. Simultaneously, each monkey moves to a neighboring vertex. A collision happens if at least two monkeys reside on the same vertex after the movement or intersect on an edge. Return the number of ways the monkeys can move so that at least one collision happens. Since the answer may be very large, return it modulo 109 + 7. Example 1: Input: n = 3 Output: 6 Explanation: There are 8 total possible movements. Two ways such that they collide at some point are: Monkey 1 moves in a clockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 2 collide. Monkey 1 moves in an anticlockwise direction; monkey 2 moves in an anticlockwise direction; monkey 3 moves in a clockwise direction. Monkeys 1 and 3 collide. Example 2: Input: n = 4 Output: 14 Constraints: 3 <= n <= 109

Explanation

Here's the solution to the monkey collision problem:

• Total Possible Moves - Non-Colliding Moves: The core idea is to calculate the total number of possible moves for all monkeys and then subtract the number of ways they can move without any collisions. The result gives us the number of ways collisions do happen.

• Total Moves: Each monkey has two choices: move clockwise or counter-clockwise. With 'n' monkeys, there are 2ⁿ total move combinations.

• Non-Colliding Moves: There are only two ways for all monkeys to move without collisions: all move clockwise, or all move counter-clockwise.

• Runtime & Storage Complexity: O(log n) runtime due to the power calculation, and O(1) storage complexity.

Code

    def monkeyMove(n: int) -> int:
    """
    Calculates the number of ways monkeys can move on a polygon such that at least one collision happens.

    Args:
        n: The number of vertices in the polygon (and monkeys).

    Returns:
        The number of collision-causing move combinations modulo 10^9 + 7.
    """
    MOD = 10**9 + 7

    # Calculate total possible moves (2^n)
    total_moves = pow(2, n, MOD)

    # Subtract the two non-colliding scenarios (all clockwise or all counter-clockwise)
    non_colliding_moves = 2
    colliding_moves = (total_moves - non_colliding_moves + MOD) % MOD  # Add MOD before % to handle negative results

    return colliding_moves