﻿ Unintentionally useful consequences of playing games with maths

# The Unintentionally Useful Consequences of Playing Games with Maths

### Christian Perfect

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...

## 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.

## 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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