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.
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.
Label this plum tree graph to make it totally magic!
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).
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?