Question 6

Take a look at different sizes of squares you can find on a chessboard. Every $2\times2$ square is going to have $2$ black squares and $2$ white squares in it. Therefore, one of the squares meeting the conditions of the question would have to be larger than $2\times2.$ Take a look at $3\times3$ squares. These squares can either have $4$ black squares (if the corners are white) or $5$ black squares (if the corners are black). Therefore, any $3\times3$ square or larger will meet the conditions of the problem.

The number of $k\times k$ squares on an $n\times n$ chessboard is $(n-k+1)^2.$ For example, on a standard $8\times8$ chessboard, there are $36$ $3\times3$ squares, $4$ $7\times7$ squares, and so on. Therefore, the answer to the problem is $\displaystyle\sum_{k=1}^6k^2=\boxed{91}$