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You heard the traveler's tales,

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you followed the crumbling maps,

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and now, after a long and dangerous quest,

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you have some good news and some bad news.

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

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containing the stash 
of ancient Stygian coins

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and the eccentric wizard 
who owns the castle

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has even generously
agreed to let you have them.

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

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

NOTE Paragraph

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The task sounds simple enough.

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Both faces of each coin bear
the fearsome scorpion crest,

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one in silver,

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one in gold.

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

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

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

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and you're left in total darkness.

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There are hundreds 
of coins in front of you

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and each one feels the same on both sides.

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

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but it's hopeless.

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You've lost track.

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But you do know one thing for certain.

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When there was still light,

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

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What can you do?

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

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You're tempted to kick the pile of coins

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and curse the curiosity 
that brought you here.

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But at the last moment, you stop yourself.

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You just realized there's 
a surprisingly easy solution.

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What is it?

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

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Answer in: 3

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Answer in: 2

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Answer in: 1

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You carefully move aside 20 coins
one by one.

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

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and then flip each one of them over.

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That's all there is to it.

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Why does such a simple solution work?

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

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

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When you take 20 coins in the darkness,

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

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have ended up in your new pile.

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But let's suppose you got seven of them.

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This means that there are thirteen
silver-facing coins left

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in the original pile.

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

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are facing gold side up.

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

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Seven gold-facing coins and
thirteen silver-facing coins

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to match the ones in the original pile.

NOTE Paragraph

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

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whether it's all of them,

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a few,

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or none at all.

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

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We know that each coin only has
two possible options.

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

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and vice versa,

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and in any combination of 20 coins,

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the number of gold-facing
and silver-facing coins

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must add up to 20.

NOTE Paragraph

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We can prove this mathematically
using algebra.

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

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

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

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its number of gold-facing coins will be

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

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When all the coins in the new pile
are flipped,

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these gold-facing coins become
silver-facing coins,

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

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The gate swings open
and you hurry away with your treasure

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before the wizard changes his mind.

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

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to determine the way 
to your next adventure.

NOTE Paragraph

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

00:04:07.350 --> 00:04:11.549
one that comes from this video 
sponsor’s excellent website.

NOTE Paragraph

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Here we have 8 arrangements of coins.

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

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

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

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which arrangements can be made all gold?

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

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on Brilliant’s website.

NOTE Paragraph

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We love Brilliant.org because the site 
gives you tools

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

00:04:42.916 --> 00:04:47.228
by breaking puzzles into smaller pieces 
or limited cases,

00:04:47.228 --> 00:04:49.648
and working your way up from there.

00:04:49.648 --> 00:04:52.928
This way, you're building up a 
framework for problem solving,

00:04:52.928 --> 00:04:55.728
instead of just memorizing formulas.

00:04:55.728 --> 00:04:59.308
You can sign up for Brilliant for free, 
and if you like riddles

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a Brilliant.org premium membership 
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00:05:02.783 --> 00:05:05.794
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Try it out today by visiting 
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