Question 4
Consider a partition of an $8\times8$ chessboard into $p$ rectangles whose interiors are disjoint such that each rectangle contains an equal number of white and black cells. Assume that $a_1\lt a_2\lt \cdots\lt a_p$, where $a_i$ denotes the number of white cells in the $i$th rectangle. Find the maximal $p$ for which such a partition is possible and for that $p$ determine all possible corresponding sequences $a_1,a_2,\ldots,a_p$.