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

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?

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.