you followed the crumbling maps,

你跟著殘缺不全的地圖。

and now, after a long and dangerous quest,

而現在，經過漫長而危險的探索，

you have some good news and some bad news.

你有一些好消息和一些壞消息。

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

好消息是你已經成功找到了傳說中的地牢。

containing the stash of ancient Stygian coins

貯藏著古代斯提亞人的錢幣。

and the eccentric wizard who owns the castle

和擁有城堡的古怪巫師。

has even generously agreed to let you have them.

甚至慷慨地同意讓你擁有它們。

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

壞消息是，他沒有那麼大度

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

讓你離開地牢，除非你解決他的謎題。

The task sounds simple enough.

這個任務聽起來很簡單。

Both faces of each coin bear the fearsome scorpion crest,

每枚硬幣的兩面都有可怕的蠍子徽章。

one in silver,

一個是銀色的。

one in gold.

一個在黃金。

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

而你要做的就是把它們抽成兩堆。

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

使每個人的硬幣數量相同，面向銀色的一面向上。

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

你正準備開始的時候，所有的火把突然都熄滅了。

and you're left in total darkness.

你就會陷入一片黑暗之中。

There are hundreds of coins in front of you

你的面前有幾百個硬幣

and each one feels the same on both sides.

而每一個人的兩邊感覺都是一樣的。

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

你試著回憶銀面幣在哪裡。

but it's hopeless.

但它是無望的。

You've lost track.

你已經失去了蹤跡。

But you do know one thing for certain.

但有一件事你是知道的

When there was still light,

當還有光亮的時候。

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

你數了數，正好有20枚銀色邊上的硬幣在堆裡。

What can you do?

你能做什麼？

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

你是否註定要和你的新發現的寶藏永遠留在地牢裡？

You're tempted to kick the pile of coins

你很想把那堆硬幣踢出去。

and curse the curiosity that brought you here.

並詛咒把你帶到這裡的好奇心。

But at the last moment, you stop yourself.

但在最後一刻，你阻止了自己。

You just realized there's a surprisingly easy solution.

你才發現有一個出人意料的簡單解決方案。

What is it?

什麼事？

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

如果你想自己想明白的話，請在這裡暫停。

Answer in: 3

答案：3

Answer in: 2

答案：2

Answer in: 1

答案：1

You carefully move aside 20 coins one by one.

你小心翼翼地把20枚硬幣一個個挪到一邊。

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

哪種無所謂：任何幣都可以。

and then flip each one of them over.

然後把他們每個人都翻過來。

That's all there is to it.

這就是全部的內容。

Why does such a simple solution work?

為什麼這麼簡單的解決方案就能成功？

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

好吧，先不管有多少硬幣。

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

重要的是，總共只有20個面向銀色的一面。

When you take 20 coins in the darkness,

當你在黑暗中拿了20個硬幣。

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

你無從得知這些銀面錢幣的數量。

have ended up in your new pile.

已經結束了在你的新堆。

But let's suppose you got seven of them.

但我們假設你有七個人。

This means that there are thirteen silver-facing coins left

這意味著還剩下十三枚銀面幣

in the original pile.

在原來的一堆。

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

這也意味著你新的那堆硬幣中的其他13個硬幣

are facing gold side up.

是面向黃金的一面。

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

那麼，當你把新的一堆硬幣都翻過來的時候，會發生什麼？

Seven gold-facing coins and thirteen silver-facing coins

7枚金面幣和13枚銀面幣。

to match the ones in the original pile.

以配合原來那一堆的。

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

事實證明，無論你搶到多少銀面幣，這都是有效的。

whether it's all of them,

是否是所有的人。

a few,

一些。

or none at all.

或根本沒有。

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

這是因為所謂的互補事件。

We know that each coin only has two possible options.

我們知道，每個硬幣只有兩種可能的選擇。

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

如果不是面向銀色的一面朝上，那一定是金色的一面朝上。

and vice versa,

反之亦然。

and in any combination of 20 coins,

並在20枚硬幣的任意組合中。

the number of gold-facing and silver-facing coins

金銀幣的數量

must add up to 20.

必須加到20。

We can prove this mathematically using algebra.

我們可以用代數在數學上證明這一點。

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

原堆中剩餘的銀面硬幣數量

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

將永遠是20減去多少你移動到新的一堆。

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

由於你的新堆也有總共20個硬幣。

its number of gold-facing coins will be

其金面幣的數量將是。

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

20減去你移動的銀面幣數量。

When all the coins in the new pile are flipped,

當新堆中的所有硬幣都被翻開時。

these gold-facing coins become silver-facing coins,

這些金面幣變成了銀面幣。

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

所以現在兩堆銀面幣的數量是一樣的。

The gate swings open and you hurry away with your treasure

大門一開，你就帶著你的寶藏匆匆離開了

before the wizard changes his mind.

在巫師改變主意之前

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

在下一個十字路口，你拋出一個你辛苦賺來的硬幣。

to determine the way to your next adventure.

來決定你下一次冒險的方向。

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

但在你走之前，我們還有一個快速的硬幣謎語給你--------。

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

其中，來自這個視頻贊助商的優秀網站。

Here we have 8 arrangements of coins.

這裡我們有8種錢幣的排列方式。

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

你可以隨意翻轉相鄰的硬幣對。

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

一翻總是金變銀，銀變金。

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

你能不能想辦法分辨，一目瞭然。

which arrangements can be made all gold?

哪些安排可以做到全金？

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

您可以嘗試這個謎題的互動版本，並確認您的解決方案

on Brilliant's website.

在Brilliant的網站上。

We love Brilliant.org because the site gives you tools

我們愛Brilliant.org，因為這個網站給你提供了工具。

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

用我們最喜歡的一種方式來解決問題------。

by breaking puzzles into smaller pieces or limited cases,

通過將謎題分解成小塊或有限的案例。

and working your way up from there.

並從那裡開始工作。

This way, you're building up a framework for problem solving,

這樣，你就建立了一個解決問題的框架。

instead of just memorizing formulas.

而不是隻記住公式。

You can sign up for Brilliant for free, and if you like riddles

你可以免費註冊Brilliant，如果你喜歡猜謎語

a Brilliant.org premium membership will get you access

成為Brilliant.org的高級會員後，您可以獲得以下服務

to countless more interactive puzzles.

到無數的互動謎題。

Try it out today by visiting brilliant.org/TedEd

今天就來試試吧，訪問 brilliant.org/TedEd。

and use that link so they know we sent you.

並使用該鏈接，使他們知道我們給你發。

The first 833 of you to visit that link

前833個訪問該鏈接的人。

will receive 20% off the annual premium subscription fee.

將獲得年度高級訂閱費八折優惠。