MathLabs

Problem 4

Consider decompositions of an 8×88\times8 chessboard into pp non-overlapping rectangles. Each rectangle has as many white squares as black squares. If aia_i is the number of white squares in the ii-th rectangle, then a1<a2<⋯<apa_1<a_2<\cdots<a_p. Find the maximum possible pp, and for this pp determine all possible sequences a1,…,apa_1,\ldots,a_p.
Step 3 of 5: The largest white count cannot be 1111 or more
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32−a7≥1+⋯+6=21  ⟹  a7≥11;a7=11⇒2a7=22∈{1×22,2×11}32-a_7\ge1+\cdots+6=21\implies a_7\ge11;\quad a_7=11\Rightarrow 2a_7=22\in\{1\times22,2\times11\}
Detailed analysis

Assume p=7p=7. Since the first six terms sum to at least 1+⋯+6=211+\cdots+6=21, we have a7=32−(a1+⋯+a6)≤11a_7=32-(a_1+\cdots+a_6)\le11; equivalently the source's useful boundary is that a7=11a_7=11 would leave the minimum first-six sum. A rectangle with 1111 white and 1111 black squares has area 2222, whose only integer dimensions are 1×221\times22 or 2×112\times11, so it cannot fit on an 8×88\times8 board. Thus a7≠11a_7\ne11 and a7≤10a_7\le10.