Counting with Symmetry

Necklaces and Burnside’s lemma

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Symmetries group patterns into orbits; orbit size × fixing symmetries = size of the group.
• Burnside’s lemma: number of distinct objects = average number of patterns each symmetry fixes.
• It counts necklaces (14), square colourings (6) and chemical isomers (13 chlorobenzenes).

Steps

1. The same necklace?
2. Orbits
3. Burnside’s lemma
4. Count the necklaces
5. Colouring a square
6. Chemistry by symmetry