What is the goal of the ladybug clock puzzle?

The puzzle asks for the probability that the number six will be the very last number on the clock to be colored red.

What does 'coloring a number red' mean in the puzzle?

A number is colored red each time the ladybug touches it during its random walk around the clock.

How are the puzzles presented?

These are monthly puzzles, with the first one presented here, part of a collaboration with mathematician Peter Winkler. Answers are discussed later.

Where can I sign up to discuss the puzzles?

You can sign up to join the Zoom calls with Peter Winkler to discuss the puzzles by visiting moath.org/mindbenders.

What is the Moath Museum's year of math about?

The Moath Museum is hosting a year of math, which includes this series of monthly puzzles in collaboration with Peter Winkler.

The ladybug clock puzzle

3Blue1Brown

Education2 min read2 min video

Jan 16, 2026|722,645 views|16,483|1,166

Mathematics three blue one brown 3 blue 1 brown 3b1b 3brown1blue 3 brown 1 blue three brown one blue

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Key Moments

TL;DR

Ladybug on a clock, probability last number colored is six.

Key Insights

A ladybug moves randomly on a clock face, visiting each number once.

The puzzle asks for the probability that the number '6' is the last to be visited.

The problem involves random walks on a circular graph.

Simulations suggest certain numbers might be more or less likely to be last.

The puzzle is part of a series curated by Peter Winkler, with solutions discussed via Zoom.

Mathematical elegance is emphasized in finding the solution.

INTRODUCTION TO THE PUZZLE

This video introduces a unique mathematical puzzle involving a ladybug on a clock face. The ladybug begins at the '12' and, each second, randomly moves to an adjacent number, either one step clockwise or one step counterclockwise. The goal is to determine the probability that the number '6' is the very last number to be colored, signifying the ladybug's final visit.

THE MECHANICS OF THE LADYBUG'S MOVEMENT

The ladybug's journey is a form of random walk on the numbers 1 through 12 arranged in a circle. Each move is a discrete step, with an equal probability of moving left or right. The process continues until every number on the clock has been visited at least once, creating a sequence of visited numbers.

THE PROBABILISTIC QUESTION

The core of the puzzle lies in understanding the probability distribution of the last number to be visited. While simulations offer visual examples, showing different numbers finishing last (e.g., '3' or '1' in the provided examples), the challenge is to rigorously calculate this probability for the number '6'.

VISUALIZING THE PROCESS

The video demonstrates the concept through simulation runs, where each visited number is colored red. These visualizations illustrate how the ladybug eventually covers all numbers on the clock. Observing multiple simulations highlights that the sequence of visited numbers and, consequently, the last number visited, can vary significantly.

COLLABORATION AND FURTHER EXPLORATION

This puzzle is presented as the first in a series of monthly mathematical puzzles curated by mathematician Peter Winkler in collaboration with 3Blue1Brown. The intention is to encourage audience engagement and independent problem-solving before revealing an elegant solution.

SEEKING AN ELEGANT SOLUTION

The video emphasizes that the puzzle is designed to be pondered, with an elegant mathematical solution awaiting discovery. Participants are invited to sign up for a Zoom call with Peter Winkler, where he will discuss the refined method for answering this type of probability problem, encouraging deeper mathematical insight.