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The fabled Mirzakhani wand is the most
powerful magical item ever created.

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And that’s why the evil wizard Moldevort
is planning to use it

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to conquer the world.

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You and Drumbledrore have finally
discovered its hiding place in this cave.

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The wand is hidden by a system
of 100 magical stones—

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including a glowing keystone—
and 100 platforms.

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If the keystone is placed 
on the correct platform,

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the wand will be revealed.

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If placed incorrectly, 
the entire cave will collapse.

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The keystone is immune to all magic,
but the other stones aren't,

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meaning you can pick them up
and cast a placement spell,

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and the platform that stone
belongs on will glow.

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Place all 99 stones correctly

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and the final platform must be 
the keystone’s correct resting place.

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You’re about to get started when one
of Moldevort’s henchmen arrives

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and irreversibly seals a random stone
to a random platform.

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If you need to place a stone that belongs
on a platform that's already occupied,

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your spell will make some random
unoccupied platform glow instead.

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What are your odds of placing the keystone
on the correct platform?

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Pause now to figure it out for yourself.
Answer in 3

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Answer in 2

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Answer in 1

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Let’s imagine we knew everything 
about this situation.

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With perfect knowledge, we could label 
the stones 1 to 100,

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based on the order we plan to place them,

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and label the platforms they belong 
on in the same way.

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We’ll label the stone
the henchmen placed as 1,

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meaning it was supposed 
to go on platform 1,

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and the keystone as 100,
belonging on platform 100.

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Of course, we don’t know 
which platform is which,

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so the numbering of the platforms
is actually invisible to us.

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There are three possibilities:

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one, that first stone was placed
randomly onto its own platform,

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in which case,
you’re guaranteed to succeed.

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Two, it was placed on the keystone’s
platform and you’re doomed to fail.

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But most likely— scenario three—
it was placed somewhere else.

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Suppose the henchman placed 
stone 1 on, say, platform 45.

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Then you’d place stone 2 
on platform 2,

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3 on 3, and so on,
until you got to stone 45.

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Its platform being taken, 
a random platform would light up.

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And here, there are three possibilities:

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If it’s platform 1, you’ll win,

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because all of the remaining stones 
will go to the correct platforms.

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If platform 100 lights up,
you lose,

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because the keystone’s spot will be taken.

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Any other platform, and you’re essentially
back where you started,

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just with 54 remaining stones 
and one on the wrong platform.

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In that scenario, let’s say the spell
tells us to place stone 45 on platform 82.

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Then we place 46 to 81 correctly,
and 82 at random.

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And here we reach the same
three possibilities:

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pedestal 1, you win, 
pedestal 100, you lose,

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any other, you continue the process.

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In other words, you’re playing a game
where you have equal chances

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to win and lose,

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and some chance to delay
the decisive moment.

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No matter how many times
this process repeats,

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you’ll inevitably either place a stone
on pedestal 1 or pedestal 100

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before you reach the keystone.

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That’s all that determines whether
you succeed or fail,

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and critically, the chances 
of those events are equal.

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This can be unintuitive,
so let’s imagine another, similar game.

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Say Drumbledrore magically generates 
numbers from 1 to 100.

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If it’s a 1, you win.

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If it’s 100, you lose.

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If it’s anything else, he picks again.

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Since the odds of winning by getting 1
are the same as losing by getting a 100,

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this is a game you’re just as likely
to win as to lose.

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It might take a while, but the delays
don’t give an advantage

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to getting a 1 before 100,
or vice versa.

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The same essential reasoning
applies to our situation.

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You’re debating whether it’s worth
risking a 50/50 chance of a cave-in

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when Drumbledrore reveals 
his secret weapon:

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a rare felush felucious potion,

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which grants extraordinary luck
for a brief period of time.

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There’s a 1 in 100 chance 
the keystone’s platform was taken

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by the first stone 
and you’ve lost already,

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but otherwise, you’ve got even odds
to win or lose.

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And right now, you’re feeling lucky.