Artwork taken from DeviantArt.
Before a sword-armed hero stands a mythological n-Hydra. It begins, graciously enough, with but one head. The hero has 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 new different heads.
Way off in the distance, stands a wizard, watching the combat through his looking glass. He is, however, very bored of watching the epic combat, since our hero can only slice one hydra-head per minute, leaving the magician with nothing but an incredible desire to sleep. Our spectator is blood-thirsty though, and randomly wakes from his naps from time to time, counting the many defeated and alive heads of the myth beast.
Let be the probability that the hero defeats the n-Hydra.
Let be the time, in minutes, since the combat began at .
Consider that the hero only takes action on integer times of , and so does the wizard.
Remember, is always a positive integer.
Level 1: Show that victory is always certain for .
Level 2: Show that victory is always certain for , although it may sound counter-intuitive.
Level 3: Show that .
Level 4: Consider that the wizard takes a full-combat nap, waking up only after the end of the n-Hydra's existence. How many dead heads is he expected to find? Display the numerical result for and find a formula for a generic value of .
Level 5: Evaluate, for a generic value of , the expected number of dead n-Hydra heads and alive heads counted by the wizard throughout integer values of . Consider that the wizard immediately wakes up and leaves if the last Hydra head is defeated.