### 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.

# Limiting Probabilities

##### Stage: 5 Challenge Level:

The question requires an understanding of the definition of matrix multiplication and of the probability formulae $$p(u\ {\rm and}\ v) = p(u)\times p(v)$$ and $$p(u\ {\rm or}\ v) = p(u)+ p(v)$$
You will want to use a calculator (or software) that will work out matrix products for you.