### Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

### Tree Graphs

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has exactly one more vertex than it has edges.

### Magic Caterpillars

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

# Simply Graphs

### Why do this problem?

Introductions to graph theory can often end up being quite dry and dusty, with lots of definitions that need to be memorised. This problem invites students to engage with the different types of graph as a pictorial representation so that they can understand why different categorisations are necessary.

### Possible approach

Hand out the cards.
"Here is a set of cards. Each card shows a representation of a graph. A graph is a collection of vertices, also called nodes, joined by edges, also called arcs. Sort the cards into groups that you think belong together in some way, and sketch the graphs together with an explanation of why you have sorted them together. See how many different ways of categorising the graphs you can come up with."

While students are working, circulate and look out for groups who have come up with collections of graphs that fit standard definitions. Here are some definitions that might be appropriate.

Bring the class together and share the categorisations they came up with. When they identify a standard categorisation, share the usual terminology with them. Perhaps hand out the definitions sheet and invite them to find examples on the cards for each definition.

### Key questions

What is it about some graphs that make them similar to others?
Can you think of different properties of the graphs?
Can you trace routes from one node to another?