Maths olympiad question · P4★★★★ Expert
Climbing out of a well: olympiad question 8
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
Two snails, A and B, start climbing from the bottom of the same well at the same time.
A climbs 5 m each day and slides back 1 m each night.
B climbs 5 m on a dry day, but on day 5, day 10, day 15 and so on (every 5 days) it rains and B climbs only 3 m;
B slides back 2 m every night.
A snail is out as soon as it reaches the top during a day.
A first reaches the top on day 17, and B on day 24.
The well's depth is a whole number of metres.
How deep is the well?
Show the hints (3)
- Hint 1: Look at A alone: it first reaches the top on day 17. What is the shallowest the well could be? The deepest?
- Hint 2: B's days come in blocks of 5 (4 dry, 1 rainy): how much does a block gain? Then find how high B gets on day 23 and on day 24.
- Hint 3: A gains 5 − 1 = 4 m a day; B gains 13 m per 5-day block. The depth is more than the highest point the day before getting out, and no more than the highest point on the day out — each snail gives a range, and only one whole number is in both.
Worked solution
Have a go first, then read the solution step by step.
- 1.A: on day 16 it gets to 15 × 4 + 5 = 65 m and is still in, so the well is deeper than 65 m; on day 17 it gets to 16 × 4 + 5 = 69 m and is out, so the well is at most 69 m. Depth: 66 to 69 m.
- 2.B: each block of 5 days gains 4 × (5 − 2) + (3 − 2) = 13 m. The first 23 days are 4 blocks and 3 more days.
- 3.After night 20, B is at 4 × 13 = 52 m; day 21 up to 57 m, back to 55 m; day 22 up to 60 m, back to 58 m; day 23 up to 63 m. So B gets to 63 m on day 23, still in; on day 24 (dry) it gets to 66 m and is out. Depth: 64 to 66 m.
- 4.The only depth in both ranges is 66 m.
- ⚠️Common mistake: using one snail only; or, for B, forgetting that day 24 is a dry day, 5 m. One snail gives just a range; you need both.
Answer: 66 m
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