# The Unintentionally Useful Consequences of Playing Games with Maths

Christian Lawson-Perfect

## About me

## Unintentionally useful maths I won't be talking about today

- Found the Fibonacci numbers in my baby's stacking cups
- Used error-correcting codes to memorise a lullaby
- Wrote an algorithm for finding graph automorphisms while extending a puzzle

## The unintentionally useful maths I *will* be talking about today

One day, I was playing about with a remarkably mathematical shape.

## Platonic solids

A *polyhedron* is a 3-dimensional shape with flat sides and straight edges.

A *Platonic solid* is a solid whose faces are all the same regular polygon, with the same number of faces meeting at each vertex.

## What can we say about a polyhedron?

- Number of edges, faces, sides.
- Shapes of the faces.
- Symmetries.

## Graphs

A *graph* is a collection of points (*vertices*), and the *edges* joining them.

It doesn't matter where the points are, or what shape the edges are, or if the edges cross.

## What can we say about a graph?

- How many vertices and edges?
- How many edges connecting each vertex?
- Does it look like a hedgehog?

## What can we say about a graph?

Can you colour the vertices so that no two neighbours have the same colour?

*OR*

Can you arrange the vertices on two sides of a line so that no edge stays on the same side of the line?

A graph with this property is called *bipartite*.

## Let's make a bipartite graph

## What can we say about a graph?

Can you draw a path which goes along every *edge* once, and ends where it started?

A graph with this property is called *Eulerian*.

## The bridges of Königsberg

## What can we say about a graph?

Can you draw a path which goes through each vertex exactly once, and ends where it started?

A graph with this property is called *Hamiltonian*.

## By the way

## The Icosian game

## Let's check if some graphs are Hamiltonian

## Let's check if some graphs are Hamiltonian

## Let's check if some graphs are Hamiltonian

## Let's check if some graphs are Hamiltonian

## Let's check if some graphs are Hamiltonian

## A Theorem

Every bipartite graph with an odd number of vertices is *non-Hamiltonian*.

## The Herschel graph

A colleague showed me the *Herschel graph*, and asked what I could do with it.

The Herschel graph is the smallest non-Hamiltonian polyhedral graph.

## The Herschel graph

## The Herschel graph

*Polyhedral*?

## Unintentionally useful maths

Graphs, groups and polyhedra turn up just about everywhere!

Mathematicians continue to ask silly questions about them.

- Social networks
- Epidemiology (!!)
- The shapes of viruses
- Cryptography