You heard the traveler's tales, you followed the crumbling maps,
听说过探险者的传说的你，依据那张皱巴巴的地图，
and now, after a long and dangerous quest, you have some good news and some bad news.
在经过悠长又危险的探险后，你有一些好消息和一些坏消息。
The good news is you've managed to locate the legendary dungeon containing the stash of ancient Stygian coin
好消息是你成功找到了传说中的的地窖，那儿存放着古希腊冥河的钱币，
and the eccentric wizard who owns the castle has even generously agreed to let you have them.
而那位拥有这座城堡的古怪巫师，竟然慷慨地同意让你带走这些钱币。
The bad news is that he's not quite as generous about letting you leave the dungeon, unless you solve his puzzle.
但是坏消息是，他却并非如此大方地让你离开地窖，除非你解开他的谜题。
The task sounds simple enough.
任务看似简单。
Both faces of each coin bear the fearsome scorpion crest, one in silver, one in gold.
钱币每一面都铸有令人生畏的蝎子饰章，一面是银的，另一面是金的。
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.
你只需要把它们分为两堆，好让每一堆中银面向上的都有相同的数目。
You're about to begin when all of the torches suddenly blow out and you're left in total darkness.
当你想开始时，突然所有的火炬都灭了，你完全陷入漆黑中。
There are hundreds of coins in front of you and each one feels the same on both sides.
你的面前堆放着上百枚钱币，每一枚的两面都有相同的质感。
You try to remember where the silver-facing coins were, but it's hopeless.
你尝试回想起银面向上的钱币的所在位置，但都徒劳无功。
You've lost track. But you do know one thing for certain.
你毫无头绪了。但是你很确定一件事儿。
When there was still light, you counted exactly 20 silver-side-up coins in the pile.
当火炬还亮着时，你准确地算到那堆钱币中有20枚银面向上的。
What can you do? Are you doomed to remain in the dungeon with your newfound treasure forever?
你可以怎么做呢？你是否注定永远地跟你新发现的宝藏留在地窖里？
You're tempted to kick the pile of coins and curse the curiosity that brought you here.
你愤怒得想踢那堆钱币一脚，以及咒骂带你到此地的好奇心。
But at the last moment, you stop yourself. You just realized there's a surprisingly easy solution. What is it?
但那一瞬间，你停了下来。你刚意识到有个相当简单的解决方案。那是什么呢？
You carefully move aside 20 coins one by one.
你小心翼翼地拿出20枚钱币。
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.
哪些钱币不重要：那堆中任何钱币都行，然后你把每一枚都翻转过来。你仅需这么做。
Why does such a simple solution work?
为什么这么简单的解决方案行得通？
Well, it doesn't matter how many coins there are to start with.
其实，开始时有多少枚钱币并不重要。
What matters is that only 20 of the total are facing silver side up.
最重要的是，全部钱币中有20枚是银面向上的。
When you take 20 coins in the darkness,
当你从黑暗中抽取20枚钱币，
you have no way of knowing how many of these silver-facing coins have ended up in your new pile.
你没办法知道那些银面向上的钱币，哪些最终属于这新的一堆钱币中。
But let's suppose you got 7 of them.
但是假设你拿到7枚银面向上的。
This means that there are 13 silver-facing coins left in the original pile.
这表示还有13枚银面向上的留在原本那一堆里。
It also means that the other 13 coins in your new pile are facing gold side up.
这也表示新的一堆里有13枚钱币是金面向上的。
So what happens when you flip all of the coins in the new pile over?
当你把新的这堆全部都翻转过来，那会发生什么事呢？

Seven gold-facing coins and 13 silver-facing coins to match the ones in the original pile.
最终会有7枚金面向上和13枚银面向上，跟原本那堆银面向上的数目一样。
It turns out this works no matter how many of the silver-facing coins you grab,
结果是不论你从原本那堆拿了多少枚银面向上的，
whether it's all of them, a few, or none at all.
要么全部银面向上的，要么一些银面向上，或完全没有。
That's because of what's known as complementary events.
那是因为“互补事件。”
We know that each coin only has two possible options.
我们知道每一枚钱币只有两种可能性。
If it's not facing silver side up, it must be gold side up, and vice versa,
如果不是银面向上，那肯定就是金面向上，反过来也一样，
and in any combination of 20 coins, the number of gold-facing and silver-facing coins must add up to 20.
任何20枚钱币的组合，金面向上和银面向上的总和一定要等于20。
We can prove this mathematically using algebra.
我们可以使用数学中的代数来证明。
The number of silver-facing coins remaining in the original pile
原本那堆中所剩下的银面向上的钱币，
will always be 20 minus however many you moved to the new pile.
永远是20减去你放到新的那堆中银面向上的数目。
And since your new pile also has a total of 20 coins,
既然新的那堆钱币的总和也是20，
its number of gold-facing coins will be 20 minus the amount of silver-facing coins you moved.
新的那堆金面向上的数目就是20减去你所抽到银面向上的数目。
When all the coins in the new pile are flipped, these gold-facing coins become silver-facing coins,
当新的那堆中所有的钱币都翻转过来，本来金面向上的就会变成银面向上的，
so now the number of silver-facing coins in both piles is the same.
所以现在银面向上的数目会跟原本那堆的一样。
The gate swings open and you hurry away with your treasure before the wizard changes his mind.
闸门打开了，你急忙地带着你的宝藏离开，以免那位巫师改变主意。
At the next crossroads, you flip one of your hard-earned coins to determine the way to your next adventure.
当你来到下一个岔口时，你掷一枚辛苦取得的钱币，决定你接下来探险旅程该走的路。
But before you go, we have another quick coin riddle for you
在你离开前，我们还有另外一个简短的钱币谜题给你，
one that comes from this video sponsor's excellent website.
这题来自这段视频赞助者的杰出的网站。
Here we have 8 arrangements of coins.
我们这儿有8种钱币组合。
You can flip over adjacent pairs of coins as many times as you like.
你可以无数次地翻转两个相邻的硬币。
A flip always changes gold to silver, and silver to gold.
每次翻转就会把金的变成银的，而银的变成金的。
Can you figure out how to tell, at a glance, which arrangements can be made all gold?
你可以看一眼就知道哪些组合可以全部变成金的吗？
You can try an interactive version of this puzzle and confirm your solution on Brilliant's website.
如果你要尝试这个互动性谜题以及确认你的答案，你可以上Brilliant的网站。
We love Brilliant.org because the site gives you tools to approach problem-solving in one of our favorite ways
我们喜爱Brilliant.org，因为这个网站给你提供工具，以我们最喜欢的方式来解决问题，
by breaking puzzles into smaller pieces or limited cases, and working your way up from there.
把谜题分解成小部分或有限的事例，然后从中慢慢解答问题。
This way, you're building up a framework for problem solving, instead of just memorizing formulas.
这样，你就是在建立一个解决问题的结构，而不是单单牢记公式。
You can sign up for Brilliant for free,
你可以免费注册Brilliant账户，
and if you like riddles, a Brilliant.org premium membership will get you access to countless more interactive puzzles.
如果你喜欢谜题，Brilliant.org的高级会员让你接触到更多互动性的谜题。
Try it out today by visiting brilliant.org/TedEd and use that link so they know we sent you.
你可以现在就通过点击以上链接浏览brilliant.org/TedEd并尝试使用，好让他们知道你是我们推荐的。
The first 833 of you to visit that link will receive 20% off the annual premium subscription fee.
前833位通过以上链接访问的人，可享有20%的每年高级会员费折扣。