# The Hydra

**Discrete Mathematics**Level 5

Before the hero stands a mythological Hydra. It begins, graciously enough, with but one head. The hero has but one way to attack the beast - slice off its head.

But this is no sure method, for, with equal probabilities, a severed head of the Hydra may either wither into nothing... or sprout again as three, new, different heads. When such an event happens, the hero has no choice but to slice again at another head, each exactly like all the others - each head having the same chance of either withering away, or sprouting into three new heads instead.

Let \(P\) be the probability that the hero defeats the Hydra - that is, he succeeds in slaying every single head, and the last head he slays does not regrow. The alternative option being that he spends an eternity in battle with the Hydra, never defeating all of its heads.

What is \(\lfloor 10^{10} P \rfloor\)?

###### Image Credit: http://powerlisting.wikia.com/wiki/Hydra*Physiology?file=Hydra*vs_Monk.jpg

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