I think you could do a mapping where white squares are considered odd numbers and black squares are considered even numbers. Now you need to figure out how to count how many odd indices there are in a matrix and then compare that to the number of even indices.
Base case is extremely easy since we can take and so the number indices altogether would be one. Thus there is 1 white square and 0 black squares, i.e., one more white than black.
Now assume there is one more white square than black squares for a board. What happens when you add two more rows and two more columns? Remember the rows and columns must remain odd.