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292. Nim Game.md

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292. Nim Game

1. Description

You are playing the following Nim Game with your friend:

  • Initially, there is a heap of stones on the table.
  • You and your friend will alternate taking turns, and you go first.
  • On each turn, the person whose turn it is will remove 1 to 3 stones from the heap.
  • The one who removes the last stone is the winner.

Given n, the number of stones in the heap, return true if you can win the game assuming both you and your friend play optimally, otherwise return false.

Example 1:

Input: n = 4
Output: false
Explanation: These are the possible outcomes:
1. You remove 1 stone. Your friend removes 3 stones, including the last stone. Your friend wins.
2. You remove 2 stones. Your friend removes 2 stones, including the last stone. Your friend wins.
3. You remove 3 stones. Your friend removes the last stone. Your friend wins.
In all outcomes, your friend wins.

Example 2:

Input: n = 1
Output: true

Example 3:

Input: n = 2
Output: true

Constraints:

  • 1 <= n <= 2^31 - 1

2. Solutions

Solution 1: Language: C Game Theory

All players are playing optimally, and the one whose amount of stones are only 1, 2 or 3 can win the game when in their turn. So if you can maintain such a scenario, your opponent has to leave 1, 2, or 3 stones no matter how they choose. Definitely, they know how to win, and they would not leave an amount to let you win unless they must do that. How is this possible?

Well, if you leave 5/6/7 stones, they could remove 1/2/3 stones and then you must do the next turn. You could remove 1/2/3 stones as well, so the opponent could win easily by removing 3/2/1 stones. To prevent this, you must leave 4 stones to win.

In conclusion, the player would be lost the game if he had $4k (k \in \mathbb{Z}^{+})$ stones at the beginning of their round because it is impossible to leave $4k$ stones when they end their round.

So...

  • Wednesday, 23 February, 2022
  • Time Complexity: $O(1)$
  • Space Complexity: $O(1)$
  • Runtime: 0 ms, faster than 100.00% of C online submissions for Nim Game.
  • Memory Usage: 5.5 MB, less than 40.97% of C online submissions for Nim Game.
bool canWinNim(int n) {
    return n % 4 != 0;
}