Die Another Day

The hydra was a many-headed monster from Greek mythology that could grow one or two new heads when one of its heads got cut off. It was the second task of Hercules to kill this monster. In an abstract sense a hydra can be modeled by a tree where the leaves are the heads, and when a head is cut off some subtrees get duplicated. Different hydra species differ by the location of subtrees to be duplicated and by the number of new subtrees grown in each step. Using some deep mathematics, it had been shown that two classes of rather restricted hydra species must always die, independent of the order in which heads are cut off. In this paper we provide an elementary proof which actually gives a complete classification of all hydra species as immortal or doomed. Now, if Hercules had known this...