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 7 of them.
This means that there are 13 silver-facing coins left
in the original pile.
It also means that the other 13 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 13 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.
This way, you're building up a framework for problem solving,
instead of just memorizing formulas.
You can sign up for Brilliant for free, and if you like riddles
a Brilliant.org premium membership will get you access
to countless more interactive puzzles.
Try it out today by visiting brilliant.org/TedEd
and use that link so they know we sent you.
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