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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.

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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.

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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: Challenge Level:2 Challenge Level:2

The Olympic emblem consists of five overlapping rings containing nine regions. In order to contribute to a pension fund for the retiring International Olympic Committee people are asked to deposit money into each region.

The guidelines allow the delegate to take all the money in any one of the rings. Place the numbers 1, 2, ... 9 in the nine regions so that the amount in each ring is the same. How many different ways can you find to do this? (Problem from University of Sydney Mathematics Enrichment Groups 1999)
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