**π‘ Problem Formulation:** We are faced with the challenge of determining whether it’s possible to fill a square matrix so that each row and column contains only distinct elements. The task is akin to solving a puzzle: given the length of a side of the square matrix, can we populate it with numbers that ensure no repeats occur in any row or column? For input, we take size n, and the output is a boolean value indicating the feasibility of such a matrix construction.

## Method 1: Brute Force Approach

This approach involves trying out all possible permutations of numbers within the matrix to check if a suitable arrangement exists that satisfies the distinctness condition for both rows and columns. The function specification is to return a boolean value after exhaustively checking all permutations.

Here’s an example:

from itertools import permutations def can_fill_square_brute_force(n): numbers = list(range(1, n+1)) perms = permutations(numbers) for grid_perm in permutations(perms, n): if all(len(set(row)) == n for row in grid_perm) and all(len(set(col)) == n for col in zip(*grid_perm)): return True return False print(can_fill_square_brute_force(3))

Output:

True

This code snippet defines a function that uses permutations from the itertools module to generate all possible arrangements of numbers from 1 to n. It then checks each permutation to see if all rows and columns contain distinct numbers, if such a permutation exists it returns True, otherwise False after all possibilities are exhausted. The drawback is that it is highly inefficient for larger values of n due to the factorial time complexity.

## Method 2: Backtracking

Backtracking is an improved recursive strategy that builds the matrix one element at a time and abandons a partial construction as soon as it violates the distinct elements condition. This method can potentially save a lot of computation time compared to brute force.

Here’s an example:

def is_safe(matrix, row, col, num): for x in range(col): if matrix[row][x] == num: return False for x in range(row): if matrix[x][col] == num: return False return True def can_fill_square_backtracking(matrix, n, col=0): if col >= n: return True for i in range(1, n+1): if is_safe(matrix, col, col, i): matrix[col][col] = i if can_fill_square_backtracking(matrix, n, col+1): return True matrix[col][col] = 0 return False n = 3 matrix = [[0]*n for _ in range(n)] print(can_fill_square_backtracking(matrix, n))

Output:

True

The example defines the function `can_fill_square_backtracking()`

which attempts to fill the matrix starting from the first column and recursively checks for safe placements of each number. If a number cannot be placed without repeating, it backtracks. It’s more efficient than brute force, but still has a significant computation cost for larger n because it explores many partial solutions before finding the correct one, if it exists.

## Method 3: Greedy Construction

The greedy construction method fills the square matrix row by row or column by column sequentially, choosing the smallest unused number at each step. It is based on the assumption that earlier choices will not prevent completion of the square.

Here’s an example:

def can_fill_square_greedy(n): if n == 1: return False matrix = [[(j+i)%n + 1 for i in range(n)] for j in range(n)] return matrix n = 3 print(can_fill_square_greedy(n))

Output:

[[1, 2, 3], [2, 3, 1], [3, 1, 2]]

This code generates a Latin square which is a special case of the desired square matrix where each number appears exactly once in each row and column. This method is very fast and guarantees a filled square for all possible n, except when n = 1, which is a trivial case that can’t be filled with distinct elements since there’s only one cell.

## Method 4: Mathematical Construction

This method uses mathematical properties to construct a Latin square or similarly structured square matrix. For instance, one might leverage algebraic structures like finite fields or orthogonal arrays to guarantee the distinctness property.

Here’s an example:

def can_fill_square_math(n): matrix = [[(i+j)%n for i in range(n)] for j in range(n)] return matrix n = 3 print(can_fill_square_math(n))

Output:

[[0, 1, 2], [1, 2, 0], [2, 0, 1]]

Here, we are constructing a Latin square using a simple modular arithmetic approach. For other matrix sizes, particularly prime or power of prime sizes, more complex algebraic structures would be required. This method can fill a matrix very efficiently, but its implementation might be complex for those unfamiliar with the required mathematical concepts.

## Bonus One-Liner Method 5: Using NumPy Library

If you’re looking for a quick solution and are comfortable using Python libraries, NumPy’s functions can be cleverly utilized to create a Latin square.

Here’s an example:

import numpy as np def can_fill_square_numpy(n): return np.indices((n, n)).sum(axis=0) % n + 1 n = 3 print(can_fill_square_numpy(n))

Output:

[[1 2 3] [2 3 1] [3 1 2]]

This one-liner takes advantage of NumPy’s array manipulation capabilities to construct a Latin square by creating a matrix of row and column indices, summing them, and then taking the result modulo n. It’s a fast and elegant solution for users who are able to install and import the NumPy library.

## Summary/Discussion

**Method 1: Brute Force Approach.**Exhaustively checks permutations. Inefficient for large n due to factorial time complexity. Not recommended for practical use.**Method 2: Backtracking.**More efficient than the brute force. Still has a high computation cost for larger n. Good for smaller sizes or when brute force is impractical.**Method 3: Greedy Construction.**Fast and ensures a result for n > 1. Produces Latin squares. May not be extendable to complex cases where distinct elements aren’t sequential numbers.**Method 4: Mathematical Construction.**Efficient. Requires knowledge of abstract algebra for complex constructions. Better suited for cases with special size constraints like prime numbers.**Bonus Method 5: Using NumPy Library.**Quick and elegant. Reliant on external libraries. Not desirable if the goal is to avoid third-party dependencies.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.