#3360

Easy

LeetCode Problem Workspace

Stone Removal Game

Alice and Bob play a game of stone removal. Alice goes first, and the winner is the player who can make a move until the other player cannot.

math simulation

Problem Statement

In the Stone Removal Game, Alice and Bob take turns removing stones from a pile. Alice starts first. On each turn, a player must remove a positive number of stones. The player who cannot make a move loses the game.

Given a positive integer n, representing the initial number of stones in the pile, return true if Alice wins the game and false otherwise. The constraints are small enough that a brute-force solution is feasible.

Examples

Example 1

Input: n = 12

Output: true

Example 2

Input: n = 1

Output: false

Constraints

1 <= n <= 50

Solution Approach

Simulate the Game

A brute-force approach involves simulating the game and tracking the turns taken by both players. By alternating between Alice and Bob, we can determine who will run out of moves first and thus who wins.

Mathematical Insight

An optimal solution might look for a pattern in the number of stones. For example, it turns out that Alice wins if n is not a multiple of 2, since she always has a move to make.

Memoization for Optimization

For larger values of n, memoization can be applied to avoid recalculating the results of the same subproblems repeatedly, improving efficiency.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the final approach. A brute-force solution would have a time complexity of O(n), while memoization or other optimized methods could reduce it to O(1) for each decision.

What Interviewers Usually Probe

Ability to simulate game mechanics
Understanding of memoization for optimization
Recognizing patterns in game theory

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the solution
Neglecting to simulate the game properly
Not recognizing simple patterns in the game mechanics

Follow-up variants

Allow different removal rules or turn patterns
Increase the upper limit of n
Change the starting player

FAQ

How do I determine if Alice wins in the Stone Removal Game?

By simulating the game or using mathematical insights like recognizing patterns based on the number of stones.

What is the time complexity of the brute-force solution for this problem?

The time complexity of a brute-force solution is O(n), as it simulates the game until one player runs out of moves.

Can memoization improve the performance of the solution?

Yes, memoization can avoid redundant calculations and improve the performance, especially if subproblems are solved multiple times.

What role does math play in the Stone Removal Game?

Mathematical patterns, such as whether n is even or odd, can provide insights into the winner, reducing the need for a full simulation.

How does the pattern in this problem help with solving similar problems?

Recognizing patterns in game mechanics, such as turn-taking and decision trees, is key to solving similar simulation and game theory problems.

terminal

Solution

Solution 1: Simulation

We simulate the game process according to the problem description until the game can no longer continue.

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class Solution:
    def canAliceWin(self, n: int) -> bool:
        x, k = 10, 0
        while n >= x:
            n -= x
            x -= 1
            k += 1
        return k % 2 == 1

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