### 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.

# Olympic Magic

##### Stage: 4 Challenge Level:

Why do this problem?
In this problem learners have to identify the mathematical features of the context and represent these features in graphical and algebraic forms. They will get practice in analysing and mathematical reasoning through using algebraic expressions and equations to find solutions. In order to check that they have found all possible solutions learners need to work systematically and check all possible cases.

Possible approach
This is essentially the same problem as W. Mates and there you will find the first steps for solving this problem. Then the problem Magic W Wrap-Up gives more ideas for solving the same problem.

Key questions
What is the sum of the whole numbers 1 to 9?
If the total in each circle is Xwhat can you say about the total of thenumbers that are in the intersections of the circles?