Solution to the ladybug clock puzzle

2026-02-16 · Source: 3Blue1Brown · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, quick

Summary

A puzzle explores the probability that the number six is the last to be colored on a 12-hour clock face by a ladybug taking random clockwise or counterclockwise steps. Initial simulations reveal that all numbers from 1 to 11 appear to be equally likely to be the last one touched, which is counterintuitive given the starting position at 12. The key insight involves reframing the problem: instead of calculating the probability from the start, one considers the moment the ladybug first touches a neighbor of the target number (e.g., 5 or 7 for the number 6). From this point, the probability of the target number being the last one colored is determined by reaching the other uncolored neighbor before touching the target itself.

Key takeaway

For a data scientist or puzzle enthusiast analyzing random walk problems, recognize that initial intuitions about probability distributions can be misleading. Your approach should consider reframing the problem by identifying intermediate states that simplify calculations, rather than attempting direct computation from the starting point. This method can reveal surprising uniform distributions in seemingly complex scenarios.

Key insights

All numbers on a clock face (excluding the start) are equally likely to be the last touched in a random walk.

Principles

• Reframing a problem can simplify its solution.
• Probabilities can be invariant under certain intermediate conditions.

Method

To find the probability of a number X being the last colored, wait until its neighbors (X-1 or X+1) are first touched, then calculate the probability of reaching the other uncolored neighbor before X.

In practice

• Simulate random walks to test intuitions.
• Break down complex probability problems into sub-problems.

Topics

• Random Walk
• Probability Puzzle
• Stochastic Processes
• Simulation Analysis
• Clock Puzzle

Best for: Research Scientist, AI Student, General Interest

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Editorial summary, takeaway, and curation by AIssential. Original article published by 3Blue1Brown.