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Magic W Wrap Up

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

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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.

Cube Net

Age 16 to 18 Challenge Level:
How many tours that visit each vertex once and only once can be traced along the edges of a cube? How many of these tours can return to the starting point thus completing a Hamiltonian Circuit?

How many different ways can the subsets of the set $\{a, b, c\}$ be arranged in a sequence so that each subset differs from the one before it by having exactly one element inserted or deleted?