1510. Stone Game IV

Description

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

 

Solutions

Solution: Dynamic Programming

• Time complexity: O(n^1.5)
• Space complexity: O(n)

 

JavaScript

/**
 * @param {number} n
 * @return {boolean}
 */
const winnerSquareGame = function (n) {
  const dp = Array.from({ length: n + 1 }, () => null);

  const removeStones = stones => {
    if (!stones) return false;
    if (dp[stones] !== null) return dp[stones];

    const maxValue = Math.floor(Math.sqrt(stones));

    for (let value = maxValue; value >= 1; value--) {
      if (removeStones(stones - value ** 2)) continue;

      dp[stones] = true;

      return true;
    }

    dp[stones] = false;

    return false;
  };

  return removeStones(n);
};