Olympiad topic · P2–P6
Tree planting
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.
Try one
★★ MediumA round pond in the park is 40 m around its edge.
A gardener plants a ring of trees along the edge, with neighbouring trees 4 m apart.
Then, between every two neighbouring trees, he puts 2 pots of flowers.
How many pots of flowers does he put out?
- Hint 1: Round a circle, is there a gap between the last tree and the first?
- Hint 2: First count the gaps round the circle — each gap is one 'between two neighbouring trees'.
- 1.The ring splits into 40 ÷ 4 = 10 gaps. Round a circle the ends join, so there are 10 trees too — nothing to add or take away.
- 2.2 pots per gap: 10 × 2 = 20 pots.
- ⚠️Common mistake: counting only 9 gaps (18 pots) and missing the gap between the last tree and the first.
Answer: 20 pots
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsOne straight road, sawing a log.
- ★★MediumFree2 question types · unlimited questionsBoth sides of a road, a loop with pots between.
- ★★★HardFree3 question types · unlimited questionsA clock's strikes, old lamp posts kept.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsA rectangle's edge with its area, or two spacings and their common multiples.
Tree planting: what it is and how to solve it
Planting trees evenly along a road or round a loop: how the number of trees and the number of gaps are linked.
For example: A 50 m path, a tree at both ends and every 10 m. How many trees? 50 ÷ 10 = 5 gaps, both ends planted: 5 + 1 = 6.
The key idea: Gaps = length ÷ spacing. A straight road planted at both ends: trees = gaps + 1; one end: trees = gaps; neither end: gaps − 1; a loop: trees = gaps. Sawing a log, a clock striking and climbing stairs work the same way.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download