Stone Removal Game

Stone Removal Game - Problem

Math Simulation Easy

Alice and Bob are playing a game where they take turns removing stones from a pile, with Alice going first.

Alice starts by removing exactly 10 stones on her first turn. For each subsequent turn, each player removes exactly 1 fewer stone than the previous opponent.

The player who cannot make a move loses the game.

Given a positive integer n, return true if Alice wins the game and false otherwise.

Input & Output

Example 1 — Small Pile

$ Input: n = 10

› Output: true

💡 Note: Alice removes 10 stones and wins immediately since Bob cannot make any move (pile is empty).

Example 2 — Medium Pile

$ Input: n = 25

› Output: false

💡 Note: Alice removes 10 (15 left), Bob removes 9 (6 left), Alice needs 8 but only 6 available. Alice loses.

Example 3 — Large Pile

$ Input: n = 100

› Output: true

💡 Note: After several rounds: Alice=10+8+6+4+2=30, Bob=9+7+5+3+1=25. Alice can make more complete moves.

Constraints

1 ≤ n ≤ 10⁵

Visualization

Tap to expand

Asked in

G Google 15 a Amazon 12 M Microsoft 8

The key insight is recognizing this as an arithmetic sequence problem. The optimal approach uses mathematical formulas to determine the winner in O(1) time instead of simulating the game. Alice wins if she can make the last valid move.

Common Approaches

✓ Mathematical Formula

⏱️ Time: O(1) Space: O(1)

The stones removed follow an arithmetic sequence: 10, 9, 8, 7, ... We can calculate how many complete rounds can be played and determine the winner mathematically without simulation.

Simulation Approach

⏱️ Time: O(√n) Space: O(1)

Track the pile size and simulate each player's turn. Alice starts with 10 stones, then each subsequent turn removes 1 fewer stone than the previous turn. Continue until a player cannot make their required move.

Mathematical Formula — Algorithm Steps

Step 1: Find the maximum number of turns k where sum ≤ n
Step 2: Calculate sum using arithmetic sequence formula
Step 3: Use binary search or direct calculation to find k
Step 4: If k is even, Bob made the last move (Alice wins), else Alice made last move (Bob wins)

Visualization

Tap to expand

Step-by-Step Walkthrough

Sequence

Stones removed: 10, 9, 8, 7, ...

Formula

Sum = k × (21 - k) ÷ 2

Winner

If k even, Alice wins; if k odd, Bob wins

Code -

solution.c — C

#include <stdio.h>
#include <math.h>
#include <stdbool.h>

bool solution(int n) {
    int discriminant = 441 - 8 * n;
    if (discriminant < 0) {
        return true;
    }
    
    int k = (int)((21 + sqrt(discriminant)) / 2);
    
    while (k > 0 && k * (21 - k) / 2 > n) {
        k -= 1;
    }
    
    return k % 2 == 0;
}

int main() {
    int n;
    scanf("%d", &n);
    bool result = solution(n);
    printf(result ? "true\n" : "false\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Direct mathematical calculation using quadratic formula

✓ Linear Growth

Space Complexity

O(1)

Only using variables for calculation

✓ Linear Space

12.5K Views

Medium Frequency

~15 min Avg. Time

234 Likes

Ln 1, Col 1

Smart Actions

💡 Explanation

AI Ready

💡 Suggestion Tab to accept Esc to dismiss

// Output will appear here after running code

Code Editor Closed

Click the red button to reopen

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Mathematical Formula — Algorithm Steps

Visualization

Code -

Time & Space Complexity

Select Compiler