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 <i>v</i>+<i>e</i> are used where <i>v</i> is the number


of vertices and <i>e</i> 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.




    


<p><img src="magic_c.gif" alt="Magic Caterpillar" width="370"
height="194" /></p>




 
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?