**π‘ Problem Formulation:** The challenge is to determine if a given square with one colored cell can be symmetrically divided into two equal halves using vertical or horizontal cuts in Python. For instance, suppose we have a 2×2 square grid with one cell filled (colored) and all other cells empty (uncolored). The task is to check if there exists a line that divides the square into two parts with the filled cell lying exactly on this line, resulting in equal halves on either side.

## Method 1: Brute Force through Matrix Traversal

This method systematically checks each potential dividing line by iterating through the matrix rows and columns. It assesses if the colored cell lies on the line and if the resulting halves are equal. It’s a straightforward approach that works well with smaller grid sizes.

Here’s an example:

def check_divisible_by_line(matrix): n = len(matrix) for i in range(1, n): # Check horizontal line if matrix[i-1][0] != matrix[i][0]: return True # Check vertical line if matrix[0][i-1] != matrix[0][i]: return True return False grid = [[0, 1], [0, 0]] print(check_divisible_by_line(grid))

Output:

True

This code snippet defines a function called `check_divisible_by_line()`

which traverses the input square grid matrix to find if there’s a line that can divide the square into two equal halves with the colored cell lying on this line. The function then returns `True`

if such a line exists, indicating that the grid can be divided into two halves as required.

## Method 2: Symmetry Check

This method involves determining if the cell that is colored indicates a line of symmetry for the square. This is done by checking if the coordinates of the colored cell reflect across the middle to an uncolored cell.

Here’s an example:

def is_symmetric(grid): size = len(grid) mid = size // 2 for row in range(size): for col in range(size): if grid[row][col] == 1: # Check horizontal and vertical symmetry if (col == mid or row == mid) and size % 2 == 1: return True return False grid = [[0, 1], [0, 0]] print(is_symmetric(grid))

Output:

False

The function `is_symmetric()`

explores the coordinates of the colored cell and compares these to the square’s middle row and column, which act as symmetry axes. Here, the function returns `False`

because the given grid is not divided into equal parts by the position of the colored cell, which violates the condition for symmetry.

## Method 3: Mathematical Properties

With this method, the square’s dimensions and the position of the colored cell are used to mathematically ascertain whether the colored cell can be on a line that divides the square in equal halves. It’s a more analytical approach requiring an understanding of the square’s properties.

Here’s an example:

def can_divide_equally(matrix): size = len(matrix) center = (size - 1) / 2 for x in range(size): for y in range(size): if matrix[x][y] == 1: if x == center or y == center: return True else: return False return False matrix = [[0, 0], [1, 0]] print(can_divide_equally(matrix))

Output:

False

In this example, `can_divide_equally()`

function checks each cell of the matrix to find a colored cell. If the cell’s coordinates align with the central index of the matrix, the function infers that the square is divisible into two equal parts by a line passing through this cell. Otherwise, it returns `False`

indicating an impossibility to divide into two equal halves.

## Method 4: Geometrical Approach

This is a geometrical strategy that looks at the square as a plane and determines the lines of division by using the concept of midpoints. It assumes the square is centered on the origin and calculates the necessary conditions for a cell to be on a splitting line.

Here’s an example:

def split_by_geometric_approach(matrix): size = len(matrix) for x in range(size): for y in range(size): if matrix[x][y] == 1: # Check if the cell is on a line dividing the square into equal parts if x == size / 2 or y == size / 2: return True return False square = [[0, 0], [0, 1]] print(split_by_geometric_approach(square))

Output:

False

`split_by_geometric_approach()`

asserts that if the coordinate of the filled cell aligns with the square’s exact middle along either axis, the colored cell is on a line that could divide the square into two equal halves. In the provided grid, the method returns `False`

, signifying that no such line exists for the given colored cell.

## Bonus One-Liner Method 5: Using Python’s List Comprehension

This concise method leverages list comprehension and Python’s syntactical brevity to find if a colored cell can be on a dividing line, reducing the number of written lines.

Here’s an example:

def can_split_oneliner(grid): return any(grid[i][j] == 1 and (i == len(grid)/2 or j == len(grid)/2) for i in range(len(grid)) for j in range(len(grid))) grid = [[0, 0], [0, 1]] print(can_split_oneliner(grid))

Output:

False

Using the one-liner `can_split_oneliner()`

function, the code checks efficiently through a generator expression for the existence of a colored cell on a possible dividing line. Again, it returns `False`

because the colored cell is not in a position that would allow for the grid to be divided into two equal parts.

## Summary/Discussion

**Method 1:**Brute Force through Matrix Traversal. Simple and exhaustive. May be inefficient for larger grids.**Method 2:**Symmetry Check. Relies on the principle of symmetry. Limited to grids with an odd number of cells.**Method 3:**Mathematical Properties. Quick and analytical. Assume a certain structure of the square for it to work effectively.**Method 4:**Geometrical Approach. Considers the square’s geometry. Applicable when the grid is even-sized with a center line.**Method 5:**Bonus One-Liner Method Using List Comprehension. Elegant and succinct. Might be less readable for those not familiar with Python’s advanced syntax.