7. [25] Suppose we have an octagon with all angles of $135^{\circ}$, and consecutive sides of alternating length 1 and $\sqrt{2}$. We draw all its sides and diagonals. Into how many regions do the segments divide the octagon? (No proof is necessary.)
Answer: 84
Solution: The easiest way to see the answer is to view the octagon as five unit squares in a cross arrangement, with four half-squares wedged at the corners. The center square is divided into 8 regions. The other 4 squares are each divided into 15 regions. The 4 half-squares are each divided into 4 regions. The answer is thus $8+4 \times 15+4 \times 4=84$