# Problem Description
This appears to be a matrix-based problem where we need to calculate the maximum possible area of contiguous characters in a matrix following certain rules. The input is a 2D matrix containing single uppercase letters, and we need to return an integer representing the maximum valid area that can be formed.

# Visual Facts
1. The image shows both invalid and valid patterns:
   - Invalid section shows 3x3 matrices with red highlighting
   - Valid section shows various matrices with blue highlighting

2. Invalid Patterns shown:
   - Pattern labeled "Not a square"
   - Pattern labeled "Not the same character"
   - Pattern labeled "Not the max square"

3. Valid Patterns shown with their max areas:
   - 3x3 matrix with max area=4
   - 3x3 matrix with max area=9
   - 6x6 matrix with max area=16
   - 6x6 matrix with max area=16

4. All valid blue-highlighted areas are square in shape

# Visual Patterns
1. Shape Rules:
   - Valid areas must form squares
   - The shape must be completely filled (no holes)

2. Character Rules:
   - All cells within a valid area must contain the same character

3. Area Calculation Rules:
   - Area is calculated as the number of cells in the valid region
   - The maximum area appears to be limited by the largest possible contiguous region of the same character

4. Invalid Conditions:
   - Cannot include different characters in the same area
   - Cannot form irregular shapes (must be square)
   - Must be the maximum possible area for that configuration

5. Size Pattern:
   - Valid areas can be 4 (2x2), 9 (3x3), or 16 (4x4)
   - These are perfect square numbers, suggesting a preference for square shapes

6. Optimization Rule:
   - When multiple valid areas exist, the solution should return the largest one
   - The maximum area of 16 appears to be achieved in multiple ways in the examples

# Solution
from typing import List

def solution(matrix: List[List[str]]) -> int:
    """
    Calculates the maximum area in a matrix according to specified rules.

    Args:
        matrix: A 2D list containing characters ('A', 'B', 'C' ...).

    Returns:
        The maximum area possible based on the given rules.
    """
    if not matrix or not matrix[0]:
        return 0

    rows = len(matrix)
    cols = len(matrix[0])
    dp = [[0]*cols for _ in range(rows)]
    
    max_side = 1 
    for i in range(rows):
        for j in range(cols):
            if i == 0 or j == 0:
                dp[i][j] = 1
            else:
                if (matrix[i][j] == matrix[i-1][j] == matrix[i][j-1] == matrix[i-1][j-1]):
                    dp[i][j] = min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1]) + 1
                else:
                    dp[i][j] = 1
            
            if dp[i][j] > max_side:
                max_side = dp[i][j]
    
    return max_side * max_side


# Test Script
def test():
    test_cases = [
        (
            [['A']],
            1
        ),
        (
            [['A','A'],
             ['A','A']],
            4
        ),
        (
            [['A','B'],
             ['C','D']],
            1
        ),
        (
            [['A','A','B'],
             ['A','A','A']],
            4
        ),
        (
            [['B','B','B','B'],
             ['B','B','B','B'],
             ['B','B','B','B']],
            9
        ),
        (
            [['A','B','C'],
             ['D','E','F'],
             ['G','H','I']],
            1
        ),
        (
            [['A','A','A','B'],
             ['A','A','A','B'],
             ['A','A','A','B'],
             ['C','C','C','C']],
            9
        ),
        (
            [['A','C','A'],
             ['A','B','B'],
             ['A','B','B']],
            4
        ),
        (
            [['A','A','A','A'],
             ['A','A','A','A'],
             ['A','A','A','A'],
             ['A','A','A','A']],
            16
        ),
        (
            [['A','A','A','A','A'],
             ['A','A','A','A','A'],
             ['A','A','A','A','A'],
             ['A','A','A','A','A'],
             ['A','A','A','A','A']],
            25
        ),
    ]

    for matrix, expected_output in test_cases:
        try:
            output = solution(matrix)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`matrix`: {matrix}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message
            
        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`matrix`: {matrix}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()