0:00:06.589,0:00:08.249
You heard the traveler's tales,

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you followed the crumbling maps,

0:00:10.029,0:00:12.899
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[br]the legendary dungeon

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containing the stash [br]of ancient Stygian coins

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and the eccentric wizard [br]who owns the castle

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has even generously[br]agreed to let you have them.

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

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

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The task sounds simple enough.

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Both faces of each coin bear[br]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[br]into two piles

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

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

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and you're left in total darkness.

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There are hundreds [br]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 [br]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 [br]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[br]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 [br]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 [br]a surprisingly easy solution.

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What is it?

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Pause here if you want to figure[br]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[br]one by one.

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It doesn't matter which ones:[br]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[br]coins there are to start with.

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What matters is that only 20[br]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[br]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[br]silver-facing coins left

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in the original pile.

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It also means that the other[br]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[br]all of the coins in the new pile over?

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Seven gold-facing coins and[br]thirteen silver-facing coins

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to match the ones in the original pile.

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

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whether it's all of them, [br]a few, or none at all.

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That's because of what's known[br]as complementary events.

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We know that each coin only has[br]two possible options.

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If it's not facing silver side up,[br]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[br]and silver-facing coins

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must add up to 20.

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We can prove this mathematically[br]using algebra.

0:03:20.927,0:03:24.498
The number of silver-facing coins[br]remaining in the original pile

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

0:03:29.167,0:03:32.517
And since your new pile also[br]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 [br]silver-facing coins you moved.

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When all the coins in the new pile[br]are flipped,

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these gold-facing coins become[br]silver-facing coins,

0:03:45.966,0:03:51.538
so now the number of silver-facing[br]coins in both piles is the same.

0:03:51.538,0:03:55.159
The gate swings open[br]and you hurry away with your treasure

0:03:55.159,0:03:57.268
before the wizard changes his mind.

0:03:57.268,0:04:00.877
At the next crossroads, you flip[br]one of your hard-earned coins

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to determine the way [br]to your next adventure.

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

0:04:07.350,0:04:11.549
one that comes from this video [br]sponsor’s excellent website.

0:04:11.549,0:04:14.010
Here we have 8 arrangements of coins.

0:04:14.010,0:04:18.790
You can flip over adjacent pairs of coins [br]as many times as you like.

0:04:18.790,0:04:22.919
A flip always changes gold to silver, [br]and silver to gold.

0:04:22.919,0:04:26.176
Can you figure out how to tell, [br]at a glance,

0:04:26.176,0:04:29.636
which arrangements can be made all gold?

0:04:29.636,0:04:33.916
You can try an interactive version of [br]this puzzle and confirm your solution

0:04:33.916,0:04:36.296
on Brilliant’s website.

0:04:36.296,0:04:39.386
We love Brilliant.org because the site [br]gives you tools

0:04:39.386,0:04:42.916
to approach problem-solving in [br]one of our favorite ways—

0:04:42.916,0:04:47.228
by breaking puzzles into smaller pieces [br]or limited cases,

0:04:47.228,0:04:49.648
and working your way up from there.

0:04:49.648,0:04:52.928
This way, you're building up a [br]framework for problem solving,

0:04:52.928,0:04:55.728
instead of just memorizing formulas.

0:04:55.728,0:04:59.308
You can sign up for Brilliant for free, [br]and if you like riddles

0:04:59.308,0:05:02.783
a Brilliant.org premium membership [br]will get you access

0:05:02.783,0:05:05.794
to countless more interactive puzzles.

0:05:05.794,0:05:09.636
Try it out today by visiting [br]brilliant.org/TedEd

0:05:09.636,0:05:12.528
and use that link so they know [br]we sent you.

0:05:12.528,0:05:15.887
The first 833 of you to visit that link

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