International Mathematical Olympiad 1974 Problem 4

Consider decompositions of an $8 \times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions:

(i) Each rectangle has as many white squares as black squares.

(ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1 < a_2 < \cdots < a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1, a_2, \ldots, a_p$.