Can you solve the dark coin riddle? - Lisa Winer

Practice more problem-solving at https://brilliant.org/TedEd/

Solution to the bonus riddle mentioned at the end: https://brilliant.org/tededcoinflip

You heard the travelers’ tales, you followed the maps, and now, you’ve finally located the dungeon containing a stash of ancient coins. The good news: the wizard who owns the castle has generously agreed to let you have the coins. The bad news: he’s not quite as generous about letting you leave the dungeon ... unless you solve his puzzle. Can you solve it and get out alive? Lisa Winer shows how.

Lesson by Lisa Winer, animation by Artrake Studio.

You heard the traveler's tales,

you followed the crumbling maps,

and now, after a long and dangerous quest,

you have some good news and some bad news.

The good news is you've managed to locate
the legendary dungeon

containing the stash
of ancient Stygian coins

and the eccentric wizard
who owns the castle

has even generously
agreed to let you have them.

The bad news is that he's not
quite as generous

about letting you leave the dungeon,
unless you solve his puzzle.

The task sounds simple enough.

Both faces of each coin bear
the fearsome scorpion crest,

one in silver,

one in gold.

And all you have to do is separate them
into two piles

so that each has the same number
of coins facing silver side up.

You're about to begin when all
of the torches suddenly blow out

and you're left in total darkness.

There are hundreds
of coins in front of you

and each one feels the same on both sides.

You try to remember
where the silver-facing coins were,

but it's hopeless.

You've lost track.

But you do know one thing for certain.

When there was still light,

you counted exactly
20 silver-side-up coins in the pile.

What can you do?

Are you doomed to remain in the dungeon
with your newfound treasure forever?

You're tempted to kick the pile of coins

and curse the curiosity
that brought you here.

But at the last moment, you stop yourself.

You just realized there's
a surprisingly easy solution.

What is it?

Pause here if you want to figure
it out for yourself.

Answer in: 3

Answer in: 2

Answer in: 1

You carefully move aside 20 coins
one by one.

It doesn't matter which ones:
any coins will do,

and then flip each one of them over.

That's all there is to it.

Why does such a simple solution work?

Well, it doesn't matter how many
coins there are to start with.

What matters is that only 20
of the total are facing silver side up.

When you take 20 coins in the darkness,

you have no way of knowing how many
of these silver-facing coins

have ended up in your new pile.

But let's suppose you got seven of them.

This means that there are thirteen
silver-facing coins left

in the original pile.

It also means that the other
thirteen coins in your new pile

are facing gold side up.

So what happens when you flip
all of the coins in the new pile over?

Seven gold-facing coins and
thirteen silver-facing coins

to match the ones in the original pile.

It turns out this works no matter how
many of the silver-facing coins you grab,

whether it's all of them,

a few,

or none at all.

That's because of what's known
as complementary events.

We know that each coin only has
two possible options.

If it's not facing silver side up,
it must be gold side up,

and vice versa,

and in any combination of 20 coins,

the number of gold-facing
and silver-facing coins

must add up to 20.

We can prove this mathematically
using algebra.

The number of silver-facing coins
remaining in the original pile

will always be 20 minus
however many you moved to the new pile.

And since your new pile also
has a total of 20 coins,

its number of gold-facing coins will be

20 minus the amount of
silver-facing coins you moved.

When all the coins in the new pile
are flipped,

these gold-facing coins become
silver-facing coins,

so now the number of silver-facing
coins in both piles is the same.

The gate swings open
and you hurry away with your treasure

before the wizard changes his mind.

At the next crossroads, you flip
one of your hard-earned coins

to determine the way
to your next adventure.

But before you go, we have another
quick coin riddle for you –

one that comes from this video
sponsor’s excellent website.

Here we have 8 arrangements of coins.

You can flip over adjacent pairs of coins
as many times as you like.

A flip always changes gold to silver,
and silver to gold.

Can you figure out how to tell,
at a glance,

which arrangements can be made all gold?

You can try an interactive version of
this puzzle and confirm your solution

on Brilliant’s website.

We love Brilliant.org because the site
gives you tools

to approach problem-solving in
one of our favorite ways—

by breaking puzzles into smaller pieces
or limited cases,

and working your way up from there.

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instead of just memorizing formulas.

You can sign up for Brilliant for free,
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will get you access

to countless more interactive puzzles.

Try it out today by visiting
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