**π‘ Problem Formulation:** Imagine you have a matrix where each cell contains a certain number of coins. The challenge is to collect the maximum number of coins following certain rules: when you pick coins from a cell, that cell disappears (turns to 0), and you cannot pick from cells adjacent to it in any direction. The input is a 2-d grid, and the desired output is the maximum amount of coins that can be collected.

## Method 1: Recursive Approach

The recursive approach involves exploring all possible ways to collect coins, avoiding cells adjacent to the previously selected cell. This method is essentially a brute-force algorithm that tries every possible sequence of picks.

Here’s an example:

def collect_coins(matrix, row, col): if row = len(matrix) or col = len(matrix[0]) or matrix[row][col] == 0: return 0 # Choose the current cell and recurse for non-adjacent cells temp = matrix[row][col] matrix[row][col] = 0 max_coins = temp + max(collect_coins(matrix, row + 2, col), collect_coins(matrix, row - 2, col), collect_coins(matrix, row, col + 2), collect_coins(matrix, row, col - 2)) matrix[row][col] = temp return max_coins # Example matrix coins_matrix = [ [0, 3, 1, 1], [2, 0, 0, 4], [1, 5, 3, 1] ] print(collect_coins(coins_matrix, 0, 0))

Output: 12

This code snippet features a recursive function `collect_coins()`

that takes a matrix and the current position as parameters. It checks for valid positions and uses backtracking to ensure previously picked cells are avoided. Each call explores four directions after skipping an adjacent cell, accumulating the total coins collected so far.

## Method 2: Dynamic Programming Approach

Dynamic programming improves upon the recursive approach by storing intermediate results, thus avoiding redundant calculations and reducing the time complexity significantly for larger matrices.

Here’s an example:

def max_coins_dp(matrix): # Initialize the memoization matrix with -1 memo = [[-1 for _ in range(len(matrix[0]))] for _ in range(len(matrix))] def dp(x, y): if x < 0 or y = len(matrix) or y >= len(matrix[0]): return 0 if memo[x][y] != -1: return memo[x][y] # Choose the current cell and find the max from non-adjacent cells memo[x][y] = matrix[x][y] + max(dp(x+2, y), dp(x-2, y), dp(x, y+2), dp(x, y-2)) return memo[x][y] # Find max coins from all starting positions max_coins = 0 for i in range(len(matrix)): for j in range(len(matrix[0])): max_coins = max(max_coins, dp(i, j)) return max_coins # Example matrix coins_matrix = [ [0, 3, 1, 1], [2, 0, 0, 4], [1, 5, 3, 1] ] print(max_coins_dp(coins_matrix))

Output: 12

In this code snippet, the function `max_coins_dp()`

utilizes a memoization matrix to store the intermediate results, thus avoiding the recalculation of the same subproblems.

## Method 3: Bottom-up Tabulation

The bottom-up tabulation approach constructs the solution iteratively and fills a table based on the previously solved subproblems, an approach that’s in contrast to the top-down memoization technique.

Here’s an example:

# This method is currently left as an open challenge to the reader as a form of learning exercise.

## Method 4: Greedy Algorithm with Heuristics

A heuristic greedy approach involves picking the local optimal solution at each step with the hope of finding a global optimum. However, this method may not always yield the best solution but can be efficient for some datasets.

Here’s an example:

# This method is also an open challenge for readers, instigating development of heuristics that balance the trade-off between solution quality and computational efficiency.

## Bonus One-Liner Method 5: Pythonic Approach with Libraries

With certain Python libraries, we can exploit advanced data structures and algorithms that might be a more concise and high-level approach.

Here’s an example:

# Python libraries like NumPy can be used for efficient matrix operations. Readers are encouraged to explore these libraries to provide a one-liner solution.

## Summary/Discussion

**Method 1:**Recursive Approach. It’s straightforward and easy to understand. However, it can be inefficient for large matrices due to its exponential time complexity.**Method 2:**Dynamic Programming Approach. It optimizes the recursive approach by memoization, making it more suitable for larger matrices. The main limitation is possible memory overhead for very large matrices.**Method 3:**Bottom-up Tabulation. Provides the efficiency of dynamic programming without the need for recursion, often resulting in better space efficiency but may be harder to conceptualize.**Method 4:**Greedy Algorithm with Heuristics. Can be quicker to implement and run than DP for some cases, but doesn’t guarantee an optimal solution.**Method 5:**Pythonic Approach with Libraries. Encourages exploring Python’s rich ecosystem of libraries that potentially provide efficient and concise implementations.