Stone Game IV

Hard

leetcode· Java· 2026-08-10Problem link ↗

MathDynamic ProgrammingMinimaxGame TheoryNim GameSprague–Grundy TheoremZero-Sum Game

Runtime

14 ms

Beats 73.25%

Memory

42.3 MB

Beats 76.82%

Problem

Alice and Bob take turns playing a game, with Alice starting first.

Initially, there are n stones in a pile. On each player's turn, that player makes a move consisting of removing any non-zero square number of stones in the pile.

Also, if a player cannot make a move, he/she loses the game.

Given a positive integer n, return true if and only if Alice wins the game otherwise return false, assuming both players play optimally.

Example 1:

Input: n = 1
Output: true
Explanation: Alice can remove 1 stone winning the game because Bob doesn't have any moves.

Example 2:

Input: n = 2
Output: false
Explanation: Alice can only remove 1 stone, after that Bob removes the last one winning the game (2 -> 1 -> 0).

Example 3:

Input: n = 4
Output: true
Explanation: n is already a perfect square, Alice can win with one move, removing 4 stones (4 -> 0).

Constraints:

1 <= n <= 10⁵

Solution

Java

import java.util.*;

class Solution {
    public boolean winnerSquareGame(int n) {
        boolean[] dp = new boolean[n + 1];
        for (int i = 1; i <= n; i++) {
            for (int k = 1; k * k <= i; k++) {
                if (!dp[i - k * k]) {
                    dp[i] = true;
                    break;
                }
            }
        }
        return dp[n];
    }
}