Problem Description

You are given a 2D matrix grid of size 3 x 3 consisting only of characters 'B' and 'W'. Character 'W' represents the white color, and character 'B' represents the black color.

Your task is to change the color of at most one cell so that the matrix has a 2 x 2 square where all cells are of the same color.

Return true if it is possible to create a 2 x 2 square of the same color, otherwise, return false.

Intuition

We aim to determine if we can form a uniform 2 x 2 square by changing the color of at most one cell in the 3 x 3 grid. To achieve this, we consider all possible 2 x 2 sections within the grid. For each section, we count how many cells are black ('B') and how many are white ('W').

If in any 2 x 2 section the number of black cells is not equal to the number of white cells, it implies there is a potential to achieve uniformity by changing the color of one cell. If such a section is found, we can immediately conclude that it is possible to form a 2 x 2 square of the same color, and the solution should return true.

If after checking all possible 2 x 2 sections no feasible section is found, we conclude that forming a uniform square is not possible and return false.

Solution Approach

The solution approach involves a straightforward enumeration of 2 x 2 squares within the given 3 x 3 grid. Here are the steps followed in the implementation:

1. Iterate Through Possible 2 x 2 Squares:
   We have four possible starting positions for 2 x 2 squares within the 3 x 3 grid. These positions are (0,0), (0,1), (1,0), and (1,1).

2. Count Cell Colors in Each 2 x 2 Square:
   For each starting position, consider the cells that form the 2 x 2 square. Count how many cells are black ('B') and how many are white ('W'). Use two counters, cnt1 for white cells and cnt2 for black cells.

3. Check for Feasibility of Uniformity:
   If for any 2 x 2 section the counts of white and black cells are not equal, it indicates that changing one cell's color can result in a uniform 2 x 2 square. If such a section is found, return true.

4. Return If No Uniform Square is Possible:
   If all possible 2 x 2 squares have equal counts of black and white cells, then forming such a square is impossible, and the function should return false.

The algorithm effectively utilizes nested loops to traverse and evaluate each potential square, ensuring a precise check with minimal operations.

Example Walkthrough

Let's walk through an example of the solution approach using the grid below:

grid = [
    ['B', 'W', 'B'],
    ['W', 'B', 'W'],
    ['B', 'W', 'B']
]

1. Identify Possible 2 x 2 Squares:

   The possible starting positions for 2 x 2 squares in the 3 x 3 grid are (0,0), (0,1), (1,0), and (1,1).

2. Examine Each 2 x 2 Square:

   a. Square starting at (0,0):

   ```
   B W
   W B
   ```

   • Count of 'B': 2
   • Count of 'W': 2
   • Since the counts are equal, uniformity can't be achieved here.

   b. Square starting at (0,1):

   ```
   W B
   B W
   ```

   • Count of 'B': 2
   • Count of 'W': 2
   • Again, the counts are equal, so one color cannot dominate.

   c. Square starting at (1,0):

   ```
   W B
   B W
   ```

   • Count of 'B': 2
   • Count of 'W': 2
   • The counts are equal here as well.

   d. Square starting at (1,1):

   ```
   B W
   W B
   ```

   • Count of 'B': 2
   • Count of 'W': 2
   • The counts are still equal, indicating no potential for uniformity.

3. Conclusion:

   Since all the examined 2 x 2 squares have equal counts of colors, altering one cell won't create a uniform square. Therefore, the function returns false.

Solution Implementation

Time and Space Complexity

The provided code has a time complexity of O(1) and a space complexity of O(1).

• Time Complexity O(1): The function evaluates a fixed-size grid (2x2 sub-grid check within a 2x2 structure inside the grid), performing a constant number of operations irrespective of the input size. The loops are limited to small, constant sizes with no dependence on a variable input size.

• Space Complexity O(1): The function uses a constant amount of additional space. Only a fixed number of variables (cnt1, cnt2, x, y, a, b) are declared, independent of the size of the grid.

Learn more about how to find time and space complexity quickly.