Maths olympiad question · P4★★★★ Expert
Climbing out of a well: olympiad question 7
An expert question on climbing out of a well for P4, with a step-by-step solution, diagrams and the answer. 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.
The question
A snail starts climbing from the bottom of a well.
On day 1 it climbs 74 cm; it tires, and each day after it climbs 3 cm less than the day before.
Every night it slides back 17 cm.
It is out as soon as it reaches the top during a day.
The depth is a whole number of centimetres. What is the deepest the well can be for the snail still to get out?
Show the hints (3)
- Hint 1: Whether it gets out depends on how high it gets each day. Does that highest point keep rising?
- Hint 2: Compare the highest points of two days in a row: how much higher (or lower) is the second? How does that compare with the 17 cm slide that night?
- Hint 3: The highest point rises only if the next day's climb is more than 17 cm. The climbs are 74, 71, 68, … cm; the last day climbing at least 17 cm is day 20.
Worked solution
Have a go first, then read the solution step by step.
- 1.The highest point on day k = the net gain of the first k − 1 days and nights + day k's climb.
- 2.Day k + 1's highest point against day k's: one more night (−17) and day k + 1's climb on top. The difference is day k + 1's climb − 17. So the highest point rises while the next day's climb is more than 17 cm, and falls once it is less.
- 3.The climbs: 74, 71, 68, … 3 cm less each day. Day 20 climbs 74 − 19 × 3 = 17 cm, exactly 17; day 21 only 14, less than 17. So the highest point peaks on day 20 (level with day 19) and only falls after.
- 4.Day 20's highest point = the first 20 days' climbs − the first 19 nights' slides = (74 + 17) × 20 ÷ 2 − 19 × 17 = 910 − 323 = 587 cm.
- 5.The snail never gets higher than 587 cm, so the deepest well is 587 cm; one centimetre more and it never gets out.
- 6.Check: highest points on days 19, 20, 21: 587, 587, 584 ✓
- ⚠️Common mistake: thinking the snail starts losing ground only on the day its climb is less than the slide, and counting one day too many; or taking off day 20's night slide too — getting out depends only on the highest point by day.
Answer: 587 cm
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