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: 11,13,15,17,25,30.

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? Prove that no matter how long they search it will be impossible to find any magic 6-legged caterpillars. What about magic caterpillars with even more legs?