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  • Maybe the fates got clumsy.

  • Maybe Poseidon had one of his angry days.

  • However it happened, the underworld is overcrowded,

  • and Zeus has ordered Hades to let some spirits out.

  • Hades arranges all the souls of the dead in a line before Cerberus.

  • When one of his three heads bites down on the soul in front of it,

  • they'll get returned to the land of the living.

  • Anyone to the left must get out of line and stay in Hades forever.

  • And everyone else shuffles forward, at which point Cerberus will feed again.

  • Each of the dog's heads has an equal chance of being the one to bite each time,

  • and no two ever bite simultaneously.

  • Unfortunately, Hades' minions forgot to tell you what was happening,

  • and by the time you show up there are only 99 souls left in line.

  • Hades looks furious and drawing attention to yourself won't end well.

  • But suddenly, time freezes, and Hermes steps out of the shadows.

  • He tells you he can instantly put you into the line,

  • and no one will realize what happened.

  • But he'll only grant his grace to someone clever enough

  • to take full advantage of it.

  • Choose the best place in line and he'll give you the spot.

  • Choose wrong, and he'll leave you to rot.

  • Which spot should you pick?

  • Pause the video to figure it out yourself.

  • Answer in 3

  • Answer in 2

  • Answer in 1

  • It's possible to calculate the exact probability of going free

  • in all 100 spots.

  • But there's a much simpler path to the solution

  • that requires surprisingly little calculation.

  • Imagine being anywhere in line.

  • Way up at the front, one of the three heads will pick someone at random,

  • and you'll move forward 1, 2, or 3 spaces.

  • Since each is equally likely,

  • your chance of survival from wherever you started

  • is the average of the chances from each of the three spaces in front of you.

  • And this is where you can find a huge shortcut.

  • Averages must be on or between the extremes of what you're averaging

  • they can never be higher than the highest value

  • or lower than the lowest.

  • So whatever your chances of survival are where you start,

  • one of the three places in front of you is at least as good,

  • and probably better.

  • This observation is incredibly powerful.

  • It means that wherever you are in line,

  • it'd be wise to trade your place for one of the three spots in front of you.

  • Let's ignore which for now and think of them as a trio

  • this trio's maximum value is better than this trio's, and so on.

  • Keep going and you'll reach the front...

  • These three spots must contain the extreme values

  • the best and worst probabilities

  • for the entire line.

  • In other words, they're all we need to consider.

  • Place 1 is bad.

  • Head one would save you, and the other two doom you forever.

  • That's just a 1 in 3 chance to escape.

  • Place 2 is better: head two is great,

  • head 3 is bad, and head 1 is ok in that it gives you another chance.

  • But place 3 is best, because head 3 saves you

  • while heads 1 and 2 both give you extra chances.

  • If you did want to consider the exact probabilities,

  • the odds of surviving in place 3 are 16 out of 27, or close to 60%.

  • The spots later in line tend to be very close to having a 50% chance of survival.

  • Why 50%?

  • Because every time Cerberus sends one soul up to be reborn,

  • he leaves 0, 1, or 2 souls in the underworld.

  • That averages out to one person staying for each one who gets freed.

  • But you can beat those odds handily with what you now know.

  • Hermes has places to be, and so do you.

  • He rewards your insight by sneaking you into the third spot.

  • And from there it'll be just a short wait to learn your ultimate fate.

Maybe the fates got clumsy.

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B1 hades line cerberus trio front chance

Can you solve the riddle and escape Hades? - Dan Finkel

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    林宜悉 posted on 2020/10/23
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