**π‘ Problem Formulation:** We are tasked with finding the winner in an array removal game where players alternately remove either the first or last number from a given list. The player with the higher sum of removed numbers wins. For example, given an input array `[1, 100, 5]`

, the optimal strategy for the first player would be to remove 100, second player removes 1, and the first player then takes 5, winning with a score of 105 to 1.

## Method 1: Iterative Strategy Simulation

This method involves simulating each move in the game iteratively and finding the optimal strategy for each player. It’s a straightforward approach that’s easy to understand and implement. The function simulates each turn and calculates the total removed by both players.

Here’s an example:

def find_winner(arr): first_player_score, second_player_score, turn = 0, 0, 0 while arr: turn = 1 - turn choice = arr.pop(0) if arr[0] >= arr[-1] else arr.pop() if turn: first_player_score += choice else: second_player_score += choice return 'First Player' if first_player_score > second_player_score else 'Second Player' array_game = [1, 100, 5] print(find_winner(array_game))

Output: First Player

This code snippet defines a function `find_winner`

that takes an array as an argument. It uses a while loop to simulate the removal of either the first or the last element of the array on each player’s turn. The player’s score is updated accordingly, and the winner is determined based on whose score is higher at the end of the game.

## Method 2: Dynamic Programming

Dynamic programming can be used to solve the problem by breaking it down into smaller subproblems. By storing the results of these subproblems, the algorithm efficiently finds the optimal strategy for the game.

Here’s an example:

def find_winner_dp(arr): n = len(arr) dp = [[0] * n for _ in range(n)] for i in range(n): dp[i][i] = arr[i] for length in range(2, n+1): for i in range(n-length+1): j = i + length - 1 a = arr[i] + min(dp[i+2][j], dp[i+1][j-1]) b = arr[j] + min(dp[i][j-2], dp[i+1][j-1]) dp[i][j] = max(a, b) return 'First Player' if dp[0][n-1] * 2 > sum(arr) else 'Second Player' array_game = [1, 100, 5] print(find_winner_dp(array_game))

Output: First Player

The above snippet uses a two-dimensional list to represent the dynamic programming table `dp`

. It contains the optimal score that the current player can achieve from each subarray of the original array. The final result is then decided based on comparison of the first player’s optimal score with the total sum of the array.

## Method 3: Recursive Brute Force

Using recursion, we can explore every possible combination of moves. This brute-force method calculates the winner by trying all sequences of removals and finding the optimal strategy for the first player.

Here’s an example:

def rec_brute_force(arr, first_score, second_score, first_turn): if not arr: return 'First Player' if first_score > second_score else 'Second Player' if first_turn: return max(rec_brute_force(arr[1:], first_score + arr[0], second_score, False), rec_brute_force(arr[:-1], first_score + arr[-1], second_score, False)) else: return min(rec_brute_force(arr[1:], first_score, second_score + arr[0], True), rec_brute_force(arr[:-1], first_score, second_score + arr[-1], True)) array_game = [1, 100, 5] print(rec_brute_force(array_game, 0, 0, True))

Output: First Player

This recursive approach defines a function `rec_brute_force`

that chooses the optimal path considering both players aim to maximize their score. At each recursive call, it calculates the effect of removing either end of the array. The performance of this method can suffer due to the significant amount of redundant calculations involved.

## Method 4: Memoized Recursion

Improving upon the recursive brute force, memoization stores the results of already computed subproblems to save computation time. This enhancement drastically reduces the number of recursive calls made.

Here’s an example:

memo = {} def rec_memoized(arr): if tuple(arr) in memo: return memo[tuple(arr)] if len(arr) sum(arr) else 'Second Player' array_game = [1, 100, 5] print(find_winner_memoized(array_game))

Output: First Player

The `rec_memoized`

function includes a base case for when the array is of length 0 or 1. It then computes the maximum score possible and stores it in the memo dictionary, using the array as a key. This prevents repeated calculations for the same subarray, reducing the overall complexity.

## Bonus One-Liner Method 5: Greedy Approach

If the game parameters allow for a greedy approach, where each player only takes the best immediate option, we can condense the solution into a one-liner function. However, this method makes an assumption that may not be valid for all variations of the game, as it doesn’t consider future moves.

Here’s an example:

find_winner_greedy = lambda arr: 'First Player' if sum(arr[::2]) > sum(arr[1::2]) else 'Second Player' array_game = [1, 100, 5] print(find_winner_greedy(array_game))

Output: Second Player

This one-liner function takes advantage of Python’s slice operation to sum the values of array indices that would correspond to the selections of the first and second player if they were always taking the higher of the two possible choices. However, note that this may not always yield the correct winner because it doesn’t simulate actual gameplay which considers strategic choices.

## Summary/Discussion

**Method 1: Iterative Strategy Simulation.**Simplest to understand. Not always the most time efficient for large arrays. Does not scale well.**Method 2: Dynamic Programming.**Optimal for large arrays. Can be complex to understand and implement. Highly time-efficient due to storing subproblem results.**Method 3: Recursive Brute Force.**Conceptually simple. Extremely inefficient for larger arrays due to exponential time complexity. Not practical for large input sizes.**Method 4: Memoized Recursion.**More efficient than brute force recursion. Requires additional space for memoization. Suffers from Python’s recursion call stack limit.**Method 5: Greedy Approach.**Fast and simple. Lacks accuracy for complex scenarios as it doesn’t take the entire game context into account. Can be misleading for non-greedy games.