Leetcode #1908: Game of Nim
In this guide, we solve Leetcode #1908 Game of Nim in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Alice and Bob take turns playing a game with Alice starting first. In this game, there are n piles of stones.
Quick Facts
- Difficulty: Medium
- Premium: Yes
- Tags: Bit Manipulation, Brainteaser, Array, Math, Dynamic Programming, Game Theory
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: piles = [1]
Output: true
Explanation: There is only one possible scenario:
- On the first turn, Alice removes one stone from the first pile. piles = [0].
- On the second turn, there are no stones left for Bob to remove. Alice wins.
Python Solution
class Solution:
def nimGame(self, piles: List[int]) -> bool:
def dfs(st):
lst = list(st)
for i, x in enumerate(lst):
for j in range(1, x + 1):
lst[i] -= j
if not dfs(tuple(lst)):
return True
lst[i] += j
return False
return dfs(tuple(piles))
Complexity
The time complexity is O(n·m) (typical). The space complexity is O(n·m) or optimized.
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.