7.40 Divide an $8 \times 8$ chessboard into $p$ rectangles, such that the rectangles satisfy the following conditions:
(1) The edges of each rectangle are along the grid lines of the chessboard;
(2) Each rectangle contains an equal number of white and black squares;
(3) If the number of white squares in the $i$-th rectangle is $a_{i}$, then $a_{1}<a_{2}<\cdots<a_{p}$. Find the maximum value of $p$ among all possible divisions, and for this maximum value of $p$, list all possible sequences $a_{1}, a_{2}, \cdots, a_{p}$.
［Solution］ From the given, we have
$$
a_{1}+a_{2}+\cdots+a_{p}=32 .
$$

From (3), we also have $a_{i} \geqslant i, i=1,2, \cdots, p$, thus from (1) we have
$$
1+2+\cdots+p \leqslant 32 \text {. }
$$

Solving (2) yields $p \leqslant 7$.
Since $1+2+3+4+5+6+7=28$, the difference with 32 is 4, so it is only necessary to distribute 4 among the terms while maintaining the increasing sequence. Therefore, all possible sequences are
(i) $1,2,3,4,5,6,11$;
(ii) $1,2,3,4,5,7,10$;
(iii) $1,2,3,4,5,8,9$;
(iv) $1,2,3,4,6,7,9$;
(v) $1,2,3,5,6,7,8$.

Since a rectangle with 22 cells will not appear in the 7 divided rectangles, (i) cannot be realized. The other four scenarios can all be realized, see the figure below.

Therefore, 7 is the maximum value of $p$, and (ii) - (v) are all possible sequences that meet the requirements of the problem.