A frog at the bottom of a dry well 20 metres deep climbs 3 metres during each day, then slips back 2 metres each night while it sleeps. It starts climbing on the morning of day 1. On which day does it first reach the rim?

The Frog That Counts Wrong

Issue 9 Warm-up Published 29 Aug 2026 By The Chalkboard Desk

The problem

A frog at the bottom of a dry well metres deep climbs metres during each day, then slips back metres each night while it sleeps. It starts climbing on the morning of day . On which day does it first reach the rim?

The frog only slips at night, and only if it has not yet escaped.
Once it reaches metres it is out and does not slide back.

Take a minute

Reveal hint 1

The net gain of metre per day is a trap: the frog leaves the well during a climb, not at the end of a night.

Reveal hint 2

Ask how far the frog is at the *start* of the day on which it escapes.

Solution

AnswerDay .

Derivation

Find the net progress per full day-night cycle.
Ask when the climb finishes the job.
Solve.
Check day explicitly.
Note the naive error.