Blurring an image can be achieved using convolution with a [kernel](https://en.wikipedia.org/wiki/Kernel_(image_processing)) (also known as a filter). One common blur kernel is the average kernel, also known as box blur kernel, where each entry has the same value, typically `1/n` where `n` is the total number of entries in the kernel. For this example, I'll use a 3x3 average kernel, thus arriving at the `1/9` constants

Here's a simple Python function that takes a square image in the form of a numpy array and returns a blurred version of it:

```python
import numpy as np

def blur_image(img):
    # Define a 3x3 average kernel
    kernel = np.array([[1/9, 1/9, 1/9],
                       [1/9, 1/9, 1/9],
                       [1/9, 1/9, 1/9]])
    
    # Get dimensions of the image
    img_dim = img.shape[0]
    
    # Create an empty output array of the same size as the input image
    blurred = np.zeros((img_dim, img_dim))
    
    # Iterate over the image, skipping the borders
    for i in range(1, img_dim-1):
        for j in range(1, img_dim-1):
            # Extract 3x3 region around the current pixel
            region = img[i-1:i+2, j-1:j+2]
            
            # Apply the kernel to the region to get the blurred pixel value
            blurred[i, j] = np.sum(region * kernel)
    
    # Return the blurred image
    return blurred

# Example usage:
# Assuming `image` is your input square numpy array
# blurred_image = blur_image(image)
```

Note that:
- This approach skips the borders of the image to avoid boundary issues with the 3x3 kernel.
- You can adjust the kernel size or values for different types of blurring effects.
- The complexity of this approach is quite high (especially for large images) due to nested loops. There are optimized techniques and algorithms (like the Fast Fourier Transform) to speed up convolutions, but they're more complex and beyond the scope of this simple implementation.
- Without padding, the resulting image will have a 1-pixel border that's not blurred. If you want to handle edge cases differently (e.g., padding or mirroring), that would require additional code.
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