Olympiad topic · P2–P4
Climbing out of a well
Rarely set directly in competition papers, but a classic for learning to think about the last step on its own. Good for the younger years.
Try one
★★ MediumA caterpillar climbs a bamboo pole 21 cm tall, starting at the bottom.
It climbs for 1 minute, going up 6 cm, then rests for 1 minute, sliding down 1 cm while it rests.
Then it climbs for a minute and rests for a minute again, over and over, always climbing at the same speed.
From leaving the bottom to reaching the top, how many minutes does the caterpillar take?
- Hint 1: One minute climbing plus one resting: in those 2 minutes, how much higher does it get?
- Hint 2: After the climb that reaches the top there is no rest. Where must it be just before that last climb?
- 1.The last climb is 6 cm, so before it the caterpillar must be at 21 − 6 = 15 cm.
- 2.Each climb-and-rest gains 6 − 1 = 5 cm.
- 3.To get to 15 cm: 15 ÷ 5 = 3 rounds of climb-and-rest.
- 4.Time: 3 × 2 + 1 = 7 minutes (the last climb has no rest after it).
- ⚠️Common mistake: 4 × 2 = 8 minutes, counting a rest after the last climb.
Answer: 7 minutes
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsSmall numbers, count day by day.
- ★★MediumFree2 question types · unlimited questionsBigger numbers, use the method.
- ★★★HardFree2 question types · unlimited questionsRests every few jumps, or a deeper well.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsRainy days climb less, or the depth worked out from two snails' days.
Climbing out of a well: what it is and how to solve it
A snail (or frog) climbs by day and slips back at night; on which day does it get out?
For example: A 9 m well; the snail climbs 5 m by day and slips 2 m at night. After the 2nd night it's at 6 m; on day 3 it climbs to 11 m, past the top: out on day 3.
The key idea: On the last day it reaches the top and is out — no slipping back — so depth ÷ net climb per day is wrong. Set aside the last day's climb, then see how many days come before it.
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