In the Land of Trees all the caterpillars have

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numbers on their feet and hips (vertices) and on

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their legs and body segments (edges) as shown on

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this 4 legged caterpillar. All the whole numbers

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from 1 to 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>v</mi><mo>+</mo><mi>e</mi></mrow></math> are used where 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>v</mi></mrow></math> is the number

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of vertices and 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>e</mi></mrow></math> is the number of edges.

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Biologists classify them by their vertex-sums. A

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vertex sum is the total of the numbers on the

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vertex and all the edges at that vertex. The

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caterpillar shown has vertex sums:

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

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





 
Show that one day a biologist may find a rare

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magic 4-legged caterpillar having the same sum at

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all its vertices and describe this creature.

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Could there be two species of magic 4-legged

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caterpillars with different numberings?

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Prove that no matter how long they search it will

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be impossible to find any magic 6-legged

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caterpillars.

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What about magic caterpillars with even more

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legs?

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