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1510 - Stone Game IV (Hard)

https://leetcode.com/problems/stone-game-iv/

Problem Statement​

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 <= 10^5

Approach 1: Dynamic Programming​

Let dp[i] be the result of the game with i stones. If it is true, it means Alice must win. If it is false, it means Bob must win. If there is any j that dp[i - j * j] make the other lose the game, then dp[i] would be true. For example, Alice can take j * j to make Bob into a losing state and end the game.

Written by @wingkwong
class Solution {
public:
    bool winnerSquareGame(int n) {
        int dp[n + 1];
        memset(dp, 0, sizeof(dp));
        for(int i = 1; i <= n; i++){
            for(int j = 1; j * j <= i; j++){
                dp[i] |= !dp[i - j * j];
            }
        }
        return dp[n];
    }
};