Solution Explanation for Game of Nim

This problem is a classic game theory problem, specifically a variation of the Nim game. The optimal strategy in Nim games is based on the nim-sum. However, a brute-force approach using Depth-First Search (DFS) with memoization is presented here, which is sufficient given the constraints of the problem (small number of piles and stones).

Approach:

The solution employs a recursive DFS function to explore all possible game states. Each recursive call represents a player's turn. The base case is when no stones remain in any pile; in this case, the current player loses. The recursive step explores all possible moves (removing stones from any non-empty pile), checking if the opponent (in the next recursive call) can be forced to lose. Memoization (using a dictionary or map) is crucial to avoid redundant calculations; the state of the game (the number of stones in each pile) is used as the key for the memoization table.

Time Complexity Analysis:

• Without Memoization: In the worst case, the number of possible game states is exponential. Let's say the maximum number of stones in a pile is m, and there are n piles. The number of possible states could be approximately O(m^n). The DFS would explore all these states.

• With Memoization: Memoization dramatically improves the time complexity. Each game state is visited only once. The number of unique game states is still bounded by O(m^n), but the time complexity becomes proportional to the number of unique states explored which is less than or equal to O(m^n). Because n (number of piles) is limited to 7 and m (maximum stones) is also limited to 7, the actual runtime is significantly faster than pure exponential time.

Space Complexity Analysis:

The space complexity is dominated by the memoization table. In the worst case, the memoization table could store O(m^n) entries, one for each unique game state. The recursion depth is also bounded by O(m*n). Therefore, the overall space complexity is O(m^n). Again, the given constraints limit this to a relatively small number.

Code Explanation (Python):

class Solution:
    def nimGame(self, piles: List[int]) -> bool:
        @cache  #This is the Python decorator for memoization
        def dfs(st):  #st represents the state of the game as a tuple
            lst = list(st)
            for i, x in enumerate(lst):
                for j in range(1, x + 1):
                    lst[i] -= j
                    if not dfs(tuple(lst)): #If the opponent loses, current player wins
                        return True
                    lst[i] += j #Backtrack after trying a move
            return False #If no winning move is found, current player loses
 
        return dfs(tuple(piles)) #Start DFS from the initial game state

The other languages (Java, C++, Go, TypeScript) follow a similar structure, using a Map or unordered_map to implement memoization. The core logic of the DFS and the memoization strategy remain the same across all languages. The variations mainly concern the syntax and the data structure used for memoization.

Note: While the provided solution is efficient enough for the given constraints, a more sophisticated solution involving the nim-sum would achieve linear time complexity, O(n), regardless of the number of stones in each pile. However, the nim-sum approach is more conceptually advanced and may be less straightforward to implement compared to this recursive approach.