# The Unintentionally Useful Consequences of Playing Games with Maths

### Christian Perfect

## About me

I did maths and further maths A-Level, then a pure maths Master's degree here at Newcastle.

I now work here in the School of Mathematics and Statistics, making a computer-based homework system, among other things.

but I also do recreational maths...

## Yes, that's a thing.

## The unplanned impact of maths

Surprisingly often, maths invented for no reason other than the fun of it turns out to be very useful.

Recently, 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 regular polygons, 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.

## What can we say about a graph?

- Number of vertices and edges.
- How many edges connecting each vertex?

## 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*.

## What can we say about a graph?

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

## 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*.

## 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.

## Unintentionally useful maths

Graph theory is used in physics, chemistry, and computing.

Group theory is used in physics and chemistry, and computing.

Polyhedra turn up just about everywhere!

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