An expert question on tree planting for P5–P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P2 to P5 papers. Sawing logs, striking clocks and lining up lamp posts are all the same problem.
The question
A school is fencing a rectangular lawn with posts: one ring round its edge, and one straight row across the middle, parallel to two of the sides, splitting it into two rectangles.
Neighbouring posts are 2 m apart; there is a post at every corner and at both ends of the middle row (those ends are posts of the ring).
The workers use 36 posts in all.
What is the largest the lawn's area can be, in square metres?
Show the hints (3)
💡Hint 1: Round the ring, posts = gaps. How many gaps does the middle row have, and how many extra posts does it use?
💡Hint 2: Let the two sides parallel to the middle row have q gaps each and the other two p gaps: the ring uses 2p + 2q posts, the row q − 1 more, so 2p + 3q = 36 + 1 = 37.
💡Hint 3: Area = p × q × 4. 2p and 3q always add to 37; the closer they are, the bigger 2p × 3q (6 × p × q). Try a few q with 3q near 37 ÷ 2.
Worked solution
Have a go first, then read the solution step by step.
1.Round the ring, posts = gaps. The middle row's ends are already counted on the ring, so it adds (its gaps − 1) posts.
2.Let the sides parallel to the middle row have q gaps and the other two p gaps (p at least 2, to fit the row). Posts = 2p + 2q + (q − 1) = 36, so 2p + 3q = 37.
3.Area = p × 2 × q × 2 = p × q × 4. With 2p + 3q fixed, the closer 2p and 3q, the bigger 2p × 3q, and so p × q; best with both near 37 ÷ 2.
4.Try the whole numbers nearby: q = 3, p = 14, p × q = 42; q = 5, p = 11, p × q = 55; q = 7, p = 8, p × q = 56; q = 9, p = 5, p × q = 45; q = 11, p = 2, p × q = 22. The largest is p = 8, q = 7: 56.
5.The lawn is 8 × 2 = 16 m by 7 × 2 = 14 m: the largest area is 16 × 14 = 224 m².
6.Check: the ring 30 posts, the row 7 − 1 = 6, 36 in all ✓
⚠️Common mistake: assuming length and width should be as close as possible, as with a plain ring. The middle row makes the q direction dearer, so more posts go to p; and the row's two ends aren't counted again.
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