Can you solve the dark coin riddle? - Lisa Winer

譯者: Lilian Chiu 審譯者: Helen Chang

You heard the traveler's tales,

你聽了旅人的故事，

you followed the crumbling maps,

你遵循著破爛的地圖，

and now, after a long and dangerous quest,

經過了驚險漫長的探索之旅後，

you have some good news and some bad news.

你有些好消息也有些壞消息。

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

好消息是，你得以 找到了傳說地牢的位置，

containing the stash of ancient Stygian coins

地牢裡藏著古老的冥河硬幣，

and the eccentric wizard who owns the castle

而擁有這座城堡的古怪巫師

has even generously agreed to let you have them.

甚至已經慷慨地同意讓你拿走它們。

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

壞消息是他沒那麼慷慨，

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

你若解不開他的謎題， 就不能離開地牢。

The task sounds simple enough.

這項任務聽起來挺簡單。

Both faces of each coin bear the fearsome scorpion crest,

每個硬幣的兩面都有 駭人的蠍子紋飾，

one in silver,

一面是銀色，

one in gold.

一面是金色。

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

你所要做的就是將硬幣分成兩堆，

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

每堆中銀色面朝上的 硬幣數量都相同。

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

你正要開始時， 突然所有的火炬都熄滅了，

and you're left in total darkness.

你身處在一片漆黑當中。

There are hundreds of coins in front of you

你面前有數百個硬幣，

and each one feels the same on both sides.

且每個硬幣的兩面摸起來都一樣。

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

你試著回憶銀色面的硬幣在哪裡，

but it's hopeless.

但毫無希望。

You've lost track.

你已經想不起來了。

But you do know one thing for certain.

但有一件事是你能確定的。

When there was still light,

當還有光時，

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

你算出在這一堆硬幣中剛好 有 20 個是銀色面朝上。

What can you do?

你能怎麼做？

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

你注定要和你新發現的寶藏 永遠困在地牢中嗎？

You're tempted to kick the pile of coins

你很想踹那堆硬幣一腳，

and curse the curiosity that brought you here.

並咒罵引導你來這裡的好奇心。

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.

你意識到有一個非常簡單的解答。

What is it?

解答是什麼？

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

如果你想自己解出，請在此暫停。

Answer in: 3

解答倒數：3。

Answer in: 2

解答倒數：2。

Answer in: 1

解答倒數：1。

You carefully move aside 20 coins one by one.

你小心地將 20 個硬幣 一一移到旁邊。

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

選哪個硬幣都無所謂： 任何硬幣都可以。

and then flip each one of them over.

然後將它們每一個都翻面。

That's all there is to it.

要做的就只有這樣。

Why does such a simple solution work?

為何如此簡單的解答行得通呢？

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

一開始有多少個硬幣並不重要。

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

重要的是全部的硬幣中 只有 20 個是銀色面朝上的。

When you take 20 coins in the darkness,

當你在黑暗中拿取 20 個硬幣，

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

你不可能知道在新選出來的這一堆

have ended up in your new pile.

硬幣中有多少是銀面朝上的。

But let's suppose you got 7 of them.

假設你拿到了 7 個銀面朝上的。

This means that there are 13 silver-facing coins left

那就表示還有 13 個 銀面朝上的硬幣

in the original pile.

留在原本的那堆硬幣中。

It also means that the other 13 coins in your new pile

那也意味著在新的那堆硬幣裡，

are facing gold side up.

有 13 個金面朝上的硬幣。

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

所以當你把新的那堆硬幣 全部翻面，會發生什麼事？

Seven gold-facing coins and 13 silver-facing coins

會有 7 個金面朝上的銀幣， 和 13 個銀面朝上的硬幣，

to match the ones in the original pile.

銀面朝上的硬幣數就和 原本那堆硬幣相符了。

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

最後發現，結果都是相同的， 不論你拿到幾個銀面朝上的硬幣，

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

不論是全部、幾個， 還是一個也沒有。

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

原因是所謂的互補事件。

We know that each coin only has two possible options.

我們已經知道每個硬幣 只有兩個可能性。

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

若不是銀面朝上， 就是一定是金面朝上。

and vice versa,

反之亦然，

and in any combination of 20 coins,

不論這 20 個硬幣是怎樣的組合，

the number of gold-facing and silver-facing coins

銀面朝上的硬幣數 和金面朝上的硬幣數

must add up to 20.

加起來一定是 20。

We can prove this mathematically using algebra.

我們可以用代數 來做數學上的證明。

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

留在原本那一堆硬幣中的 銀面朝上硬幣數目

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

一定會等於 20 減去被拿到 新的那堆的銀面朝上硬幣數目。

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

既然新的那堆硬幣中的 硬幣總數也是 20，

its number of gold-facing coins will be

在這堆硬幣中金面朝上的 硬幣數目就會是

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

20 減去被你移動的 銀面朝上硬幣數目。

When all the coins in the new pile are flipped,

當新的這堆硬幣中 所有硬幣都翻了一面，

these gold-facing coins become silver-facing coins,

金面朝上的硬幣 就變成了銀面朝上，

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

現在，兩堆硬幣中銀面朝上的 硬幣數目就相同了。

The gate swings open and you hurry away with your treasure

大門被打開了， 你在女巫改變心意之前

before the wizard changes his mind.

拿著寶藏立刻離開。

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

在下個十字路口，你投擲了 其中一個得來不易的硬幣，

to determine the way to your next adventure.

來決定你下次探險的方向。

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

但是在你離開前，我們還為你 準備了另一個硬幣謎題——

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

這個謎題來自這支影片 贊助商的傑出網站。

Here we have 8 arrangements of coins.

這裡我們有 8 種不同的硬幣排列。

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

你可以翻轉相鄰的一對硬幣， 翻轉次數不限。

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

翻轉就會把金色轉為銀色， 銀色轉為金色。

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

你能不能想出，要如何只看一眼，

which arrangements can be made all gold?

就知道哪種排列最後 可以轉為全部金色？

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

你可以在 Brilliant 的網站上 玩這個遊戲的互動版，

on Brilliant’s website.

來確認你的答案對不對。

We love Brilliant.org because the site gives you tools

我們很喜歡 Brilliant.org， 是因為它能提供你工具，

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

讓我們能用我們用最喜歡的 方式之一來解決問題——

by breaking puzzles into smaller pieces or limited cases,

將謎題拆散成許多小部分 或是有限的案例，

and working your way up from there.

從那裡開始一步步地破解。

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

這麼一來，你就是在建構 一個解決問題的架構，

instead of just memorizing formulas.

而不只是在背公式。

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

你可以免費註冊 Brilliant ， 如果你喜歡解謎，

a Brilliant.org premium membership will get you access

Brilliant.org 還有尊榮 會員會籍，讓你能取得

to countless more interactive puzzles.

數不清的互動式謎題。

Try it out today by visiting brilliant.org/TedEd

今天就來試用看看， 造訪 brilliant.org/TedEd，

and use that link so they know we sent you.

請用上述連結，他們才知道 是從我們這連過去的。

The first 833 of you to visit that link

最先連結過去的前 833 人

will receive 20% off the annual premium subscription fee.

享有尊榮會籍年費八折的優惠。