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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.

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

Age 14 to 18 Challenge Level:

In the Land of Trees all the caterpillars have numbers on their feet and hips (vertices) and on their legs and body segments (edges) as shown on this 4 legged caterpillar. All the whole numbers from 1 to $v+e$ are used where $v$ is the number of vertices and $e$ is the number of edges. Biologists classify them by their vertex-sums.

A vertex sum is the total of the numbers on the vertex and all the edges at that vertex.

The caterpillar shown has vertex sums:

11 (8+3), 13 (9+4), 15 (10+5), 17 (11+6), 25 (8+9+7+1) and 30 (7+10+11+2).

Magic Caterpillar

Show that one day a biologist may find a rare magic 4-legged caterpillar having the same sum at all its vertices and describe this creature.


Could there be two species of magic 4-legged caterpillars with different numberings?


Do magic 6-legged caterpillars exist?


What about magic caterpillars with even more legs?