Generating an irregular checkboard pattern in black and white requires a method that deviates from a perfectly regular grid.  There's no single "correct" answer as "irregular" is subjective.  However, here are a few approaches to create such a pattern, focusing on algorithmic generation rather than a specific visual representation (as that would require a graphics program):

**Method 1: Randomly Varying Cell Sizes**

This method starts with a base grid, but randomly adjusts the size of each cell within a defined range.  This creates a pattern where the checks are not uniformly sized.

* **Algorithm:**
    1. Define a base grid size (e.g., 10x10 cells).
    2. For each cell, generate random values for its width and height within a specified range (e.g., 80-120% of the base cell size).
    3. Assign black or white to each cell based on its position (e.g., alternating colors).  You could also introduce randomness here, but maintaining a general checkerboard feel requires careful control.
    4. Render the pattern based on the calculated cell sizes and colors.

**Method 2:  Jittered Grid**

This method uses a grid, but randomly displaces the position of each cell's corner points.

* **Algorithm:**
    1. Define a base grid size and cell size.
    2. For each cell's corner points, generate small random offsets (x and y coordinates) within a defined range.
    3. Assign black or white to each cell based on its position (alternating or random, as before).
    4. Render the pattern using the jittered corner points to define the cell boundaries.

**Method 3: Perlin Noise**

This method uses Perlin noise to generate a more organic, less grid-like pattern.

* **Algorithm:**
    1. Generate a 2D Perlin noise field.
    2. Threshold the noise values.  Values above a certain threshold are assigned black, and values below are assigned white.  Adjusting the threshold controls the density of black and white areas.
    3. This will create a more natural, less structured pattern than a grid-based approach.  The "check" size will be determined by the frequency of the Perlin noise.

**Method 4: Voronoi Diagram**

This method uses a Voronoi diagram to create a pattern with irregular cells.

* **Algorithm:**
    1. Generate a set of random points.
    2. Create a Voronoi diagram from these points.  Each cell in the diagram will be a polygon.
    3. Assign black or white to each cell based on its position (alternating or random).
    4. Render the pattern.


**Implementation Notes:**

These algorithms can be implemented using various programming languages and libraries.  Libraries like Processing (Java), p5.js (JavaScript), or Python with libraries like NumPy and Matplotlib are well-suited for this task.  The specific implementation details will depend on the chosen language and library.  Consider using a graphics library to render the final image.  The level of irregularity can be controlled by adjusting parameters like the range of random values, the threshold for Perlin noise, or the density of points in the Voronoi diagram.