Tie random ends: How many loops?

2026-05-21 · Source: 3Blue1Brown · Field: Science & Research — Mathematics & Computational Sciences · Depth: Intermediate, quick

Summary

The latest "Puzzle of the Month" introduces a combinatorial probability challenge centered on forming loops from string pieces. The scenario begins with a specified number of individual strings, from which two random free ends are repeatedly selected and tied together. This iterative process continues until all available ends are connected, resulting in a collection of closed loops. The formation of a loop occurs when two ends of the same original string are tied. For instance, starting with 10 strings typically yields three loops. The specific problem asks: if one commences with 50 distinct strings and performs this random end-tying procedure 50 times, what is the average number of loops expected to be present in the final configuration?

Key takeaway

For problem solvers interested in combinatorial probability, this puzzle offers a concise challenge to apply statistical reasoning. You should consider how the number of strings and tying operations influence the expected outcome, potentially modeling the process as a graph problem or using probabilistic methods. This exercise can sharpen your intuition for random processes and expected values.

Key insights

A puzzle explores the average number of loops formed by randomly tying string ends.

Topics

• Combinatorial Probability
• Mathematical Puzzles
• Expected Value
• Graph Theory
• Random Processes

Best for: Research Scientist, General Interest

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