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Leetcode 877. Stone Game

Problem Statement

Alice and Bob play a game with piles of stones. There are an even number of piles arranged in a row, and each pile has a positive integer number of stones piles[i].

The objective of the game is to determine if Alice will always win if both Alice and Bob play optimally. Alice always goes first.

Both players take turns taking a pile of stones from either the beginning or the end of the row. The game continues until there are no more piles left. The player with the most stones wins.

Given an array piles where piles[i] is the number of stones in the i-th pile, return true if Alice wins the game, or false if Bob wins the game. The game will always have an even number of piles, i.e., piles.length is always even.

Clarifying Questions

  1. Can both players end up with the same number of stones?
    • It’s guaranteed by the problem’s constraints that Alice will win if they both play optimally.
  2. Is the number of piles always guaranteed to be even and non-empty?
    • Yes, according to the problem statement, there is always an even number of piles, and the number of stones in each pile is always a positive integer.

Strategy

Given the constraints and the nature of the problem (both players playing optimally), a direct analysis reveals that Alice can always win if both play optimally when the number of piles is even.

Key Observations

  1. Constraints: The number of piles is always even.
  2. Optimal Play: We need to consider the best possible outcome for Alice if she starts first.

Approach

  1. Simplification: Since the number of piles is even, Alice can always make moves such that she gains the advantage.
  2. Dynamic Programming would typically be a solution to games of choices with optimal strategy. However, in this particular problem, a simplified argument can be made given the problem’s constraints.
  3. Mathematical Induction can be used to prove Alice wins for any n piles when n is even.

For this problem, we can directly return true since Alice will always win by following an optimal strategy.

Code

class Solution {
public:
    bool stoneGame(vector<int>& piles) {
        // Alice can always win by playing optimally.
        return true;
    }
};

Time Complexity

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