### 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.

# Plum Tree

##### Stage: 4 and 5 Challenge Level:

There are many possible magic labellings and it is possible to work systematically and to use simple algebraic arguments to find them all (see the hint to get started). Also it is possible to find a magic vertex labelling and a magic edge labelling for 20. Jo sent us her labelled trees:

I thought about what conditions I had, and used them to narrow down the number of possibilities, then just tried some things out, and they worked! I expect that there are lots more possibilities, but I don't know. Anyway, here are my trees.

Vertex magic (each with sum $21$):

Edge magic (each with sum $20$):