11. II 5 (BUL 1) ${ }^{\mathrm{IMO} 4}$ Consider a partition of an $8 \times 8$ chessboard into $p$ rectangles whose interiors are disjoint such that each of them has an equal number of white and black cells. Assume that $a_{1}<a_{2}<\cdots<a_{p}$, 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 $a_{1}, a_{2}, \ldots, a_{p}$.
11. First, we notice that $a_{1}+a_{2}+\cdots+a_{p}=32$. The numbers $a_{i}$ are distinct, and consequently $a_{i} \geq i$ and $a_{1}+\cdots+a_{p} \geq p(p+1) / 2$. Therefore $p \leq 7$. The number 32 can be represented as a sum of 7 mutually distinct positive integers in the following ways: (1) $32=1+2+3+4+5+6+11$; (2) $32=1+2+3+4+5+7+10$; (3) $32=1+2+3+4+5+8+9$; (4) $32=1+2+3+4+6+7+9$; (5) $32=1+2+3+5+6+7+8$. The case (1) is eliminated because there is no rectangle with 22 cells on an $8 \times 8$ chessboard. In the other cases the partitions are realized as below. ![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-408.jpg?height=239&width=250&top_left_y=1281&top_left_x=234)  Case (2) ![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-408.jpg?height=239&width=243&top_left_y=1281&top_left_x=518)  Case (3) ![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-408.jpg?height=236&width=241&top_left_y=1282&top_left_x=808)  Case (4) ![](https://cdn.mathpix.com/cropped/2024_11_18_8e985d6b9c83aa3e9d0eg-408.jpg?height=238&width=243&top_left_y=1279&top_left_x=1098)  Case (5)