stone-game-iv

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1510. Stone Game IV (Hard)

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

Related Topics

[Math] [Dynamic Programming] [Game Theory]

Similar Questions

1. Stone Game V (Hard)
2. Stone Game VI (Medium)
3. Stone Game VII (Medium)
4. Stone Game VIII (Hard)
5. Stone Game IX (Medium)

Hints

Hint 1

Use dynamic programming to keep track of winning and losing states. Given some number of stones, Alice can win if she can force Bob onto a losing state.