The Hydra
Having completed 11 of the 12 labours, Hercules finally faces the last and most dangerous labour, the Hydra. The only attack Hercules has for the beast is to sever its head off.
However, every time the Hydra's head is cut, there is equal chance of it remaining beheaded or sprouting 5 monstrous heads.
Fortunately for the hero, the Hydra has only 1 head to start with. Assuming the probability that Hercules finally defeats this creature of darkness be . Find the value of
- I am terribly sorry for having almost completely duplicated this problem. It as never my intention and I did not know about the problem. Sorry.
The answer is 5187900636.
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1 solution
This is the cheaters solution :
Let P be the probability that the HYDRA is killed then we can set up an equation: P = 2 1 + 2 1 P 5 This simply means that there is one half probability that HYDRA dies addes to one half chance that P reccurs to the fifth power.
Solving the equation we see that the only root between 0 and 1 is P = 0 . 5 1 8 7 9 0 0 6 3 6 7 . . . . . . . . . Thus ⌊ 1 0 1 0 P ⌋ = 5 1 8 7 9 0 0 6 3 6
P.s. Terribly sorry for giving the exact solution from the original problem posted nearly a year ago..... Great problem and solution by the problem poster Daniel Ploch
In tribute a link to that problem where he posted a beautifully detailed combinatorial solution
The Hydra