A hard question on climbing out of a well for P3–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 frog at the bottom of a well 17 m deep wants to get out.
Each jump takes it up 2 m.
After every 3 jumps it has to rest, sliding back 1 m while it rests, and then it jumps on.
As soon as it reaches the top, it is out.
How many jumps does the frog make to get out?
Show the hints (3)
💡Hint 1: Treat 'three jumps, then a slide' as one round. How much does a round gain?
💡Hint 2: The last round may need fewer than 3 jumps, and there is no slide after getting out. How high must the frog be when its last round starts?
💡Hint 3: After 3 rounds the frog is at 15 m; from there each jump adds 2 m.
Worked solution
Have a go first, then read the solution step by step.
1.A round of 3 jumps gains 3 × 2 = 6 m, minus the 1 m slide: 5 m.
2.The last round can climb at most 6 m, so it must start at 17 − 6 = 11 m or higher.
3.11 ÷ 5 = 2 remainder 1, so 3 full rounds come first.
4.After 3 rounds it is at 3 × 5 = 15 m, 2 m short, so 1 more jump (1 × 2 = 2 m).
5.In all: 3 × 3 + 1 = 10 jumps.
⚠️Common mistake: dividing the depth by a round's net gain, forgetting the last round stops at the top with no slide.
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