Given an integer **N**, which is a power of 2, return a square of length N filled with the numbers {1, 2, 3, ... N^{2}} following the "Z Pattern".

This consists of recursively traversing the 4 quadrants in the order: upper-left, upper-right, lower-left, lower-right.

**Example 1:**

Input:N= 2Output:[ [1, 2], [3, 4] ]

**Example 2:**

Input:N= 4Output:[ [1, 2, 5, 6], [3, 4, 7, 8], [9, 10, 13, 14] [11, 12, 15, 16] ]

The core challenge of this problem is to fill a 2D array in a specific "Z Pattern" order. This pattern involves recursively dividing the matrix into four quadrants and filling them in a specific sequence. This problem is significant in understanding recursive algorithms and matrix manipulation, which have applications in computer graphics, image processing, and more.

Potential pitfalls include misunderstanding the recursive nature of the problem and incorrectly traversing the quadrants.

To solve this problem, we need to think recursively. The naive approach would be to try and fill the matrix in a single pass, but this would be complex and error-prone. Instead, we can use a divide-and-conquer strategy:

- Divide the matrix into four quadrants.
- Recursively fill each quadrant in the order: upper-left, upper-right, lower-left, lower-right.

This approach ensures that the numbers are filled in the correct "Z Pattern" order.

Here is a step-by-step breakdown of the algorithm:

- Initialize a 2D array of size N x N.
- Define a recursive function to fill the matrix.
- In the recursive function, if the current quadrant size is 1, fill it with the current number and return.
- Otherwise, divide the current quadrant into four smaller quadrants and recursively fill them in the order: upper-left, upper-right, lower-left, lower-right.

```
public class ZPattern {
public static void main(String[] args) {
int N = 4; // Example input
int[][] result = generateZPattern(N);
for (int[] row : result) {
for (int num : row) {
System.out.print(num + " ");
}
System.out.println();
}
}
public static int[][] generateZPattern(int N) {
int[][] matrix = new int[N][N];
fillZPattern(matrix, 0, 0, N, 1);
return matrix;
}
private static int fillZPattern(int[][] matrix, int row, int col, int size, int startNum) {
if (size == 1) {
matrix[row][col] = startNum;
return startNum + 1;
}
int halfSize = size / 2;
int num = startNum;
// Upper-left quadrant
num = fillZPattern(matrix, row, col, halfSize, num);
// Upper-right quadrant
num = fillZPattern(matrix, row, col + halfSize, halfSize, num);
// Lower-left quadrant
num = fillZPattern(matrix, row + halfSize, col, halfSize, num);
// Lower-right quadrant
num = fillZPattern(matrix, row + halfSize, col + halfSize, halfSize, num);
return num;
}
}
```

The time complexity of this approach is O(N^2) because we are filling each element of the N x N matrix exactly once. The space complexity is O(N^2) for storing the matrix.

Potential edge cases include:

- N = 1: The smallest possible matrix.
- N = 2: The smallest power of 2 greater than 1.

These cases are handled by the base case of the recursive function.

To test the solution comprehensively, we should include a variety of test cases:

- Simple cases like N = 1 and N = 2.
- Larger cases like N = 4, N = 8, etc.

We can use standard testing frameworks like JUnit for automated testing.

When approaching such problems, it's essential to break down the problem into smaller sub-problems and solve them recursively. Practice similar problems to develop a strong understanding of recursive algorithms and matrix manipulation.

In this blog post, we discussed how to solve the "Z Pattern" problem using a recursive approach. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills.