has been exposed for his corrupt dealings.

過去骯髒的交易被爆料出來。

Nearly all of his riches

他幾乎把所有的財富都投資到

are invested in a collection

一系列共三十顆精緻 緬甸紅寶石的收藏品上，

of 30 exquisite Burmese rubies,

而廣場上的群眾喧鬧著

and the crowd in the square

要將這些紅寶石沒收 來償還他欠受害者的錢。

is clamoring for their confiscation

但這個惡棍和他在法庭上的盟友

to reimburse his victims.

提出了很有說服力的論據，

But the scoundrel and his allies at court

說明他的財富至少 有一部分是合法取得，

have made a convincing case

並且是為君王效力而得。

that at least some of his wealth

國王沉思了一會兒， 宣佈了他的判決。

was obtained legitimately,

因為不可能知道 有多少比例的紅寶石

and through good service to the crown.

是用非法取得之財富所購買，

The king ponders for a minute

罰金將要透過 一個益智比賽來決定，

and announces his judgment.

比賽是由這名商人對抗 國王最聰明的顧問——你。

Because there's no way to know

兩位在賽前都已得知遊戲規則。

which portion of the rubies were bought with ill-gotten wealth,

商人被允許可以私底下 將他的紅寶石

the fine will be determined through a game of wits between the merchant

分別放入三個盒子中， 接著盒子會被放在你面前。

and the king's most clever advisor – you.

你會拿到三張卡片，

You're both told the rules in advance.

你必須要在每張卡片上寫下 1 到 30 之間的一個數字，

The merchant will be allowed to discreetly divide his rubies

接著在每個盒子前面 各放一張卡片。

among three boxes, which will then be placed in front of you.

再來盒子會被打開。

You will be given three cards,

你能從每個盒子得到的紅寶石數目

and must write a number between 1 and 30 on each,

就是你寫在對應卡片上的數字。

before putting a card in front of each of the boxes.

前提是那個盒子中的 紅寶石要足夠。

The boxes will then all be opened.

但如果你的數字比盒子中 實際的紅寶石數目還多，

For each box, you will receive exactly

這個惡棍就可以保住整個盒子。

as many rubies as the number written on the corresponding card,

國王只設定了兩個條件來限制

if the box has that many.

惡棍要如何分配他的紅寶石。

But if your number is greater than the number of rubies actually there,

每個盒子要有 2 顆以上的紅寶石，

the scoundrel gets to keep the entire box.

其中一個盒子中的紅寶石數目必須 比另一個盒子多出 6 顆整——

The king puts just two constraints on how the scoundrel distributes his rubies.

但你不知道是哪個盒子。

Each box must contain at least two rubies

在深思熟慮了幾分鐘之後，

and one of the boxes must contain exactly six more rubies than another—

商人把寶石藏起來， 接著盒子被送到你面前。

but you won't know which boxes those are.

你要選擇哪些數字

After a few minutes of deliberation,

才能確保讓這個惡棍 被罰最高的罰金，

the merchant hides the gems, and the boxes are brought in front of you.

讓他的受害者得到最高額的補償？

Which numbers should you choose

若你想要嘗試自己解題， 請在這裡暫停。

in order to guarantee the largest possible fine for the scoundrel

答案即將公佈：3

and the greatest compensation for his victims?

答案即將公佈：2

Pause the video now if you want to figure it out for yourself.

答案即將公佈：1

Answer in 3

你不會想要因為太貪心 而讓數字超過。

Answer in 2

但有一個方法，可以確保

Answer in 1

這個惡棍必須要交出 他至少一半的藏匿物。

You don't want to overshoot by being too greedy.

這個情況就類似 對抗遊戲，如西洋棋——

But there is a way you can guarantee

只是在這裡， 你看不到對手的位置。

to get more than half of the scoundrel's stash.

若要得知你最少一定能 得到的紅寶石數目，

The situation resembles an adversarial game like chess –

你得要去探究最糟的情境，

only here you can't see the opponent's position.

也就是當作商人 已經知道你要怎麼做，

To figure out the minimum number of rubies you're guaranteed to win,

他能夠安排紅寶石的分配， 來讓你贏得最少。

you need to look for the worst case scenario,

因為你不可能知道哪個盒子 有比較多／少紅寶石，

as if the merchant already knew your move

你應該對所有盒子 都選一樣的數字。

and could arrange the rubies to minimize your winnings.

假設你寫了三個 9。

Because you have no way of knowing which boxes will have more or fewer rubies,

這個惡棍可以把紅寶石 分配成 8 顆、14 顆、8 顆。

you should pick the same number for each.

這麼一來，你只會從中間的盒子 得到 9 顆紅寶石，沒有其他的。

Suppose you write three 9's.

另一方面，你能確定 至少有兩個盒子

The scoundrel might have allocated the rubies as 8, 14 and 8.

裡面有至少 8 顆紅寶石。

In that case, you'd receive 9 from the middle box and no others.

原因如下。

On the other hand, you can be sure that at least two boxes

我們先從相反的假設開始，

have a minimum of 8 rubies.

兩個盒子中的紅寶石 數目只有 7 顆或以下。

Here's why.

這兩個盒子的寶石數目 不可能相差 6 顆，

We'll start by assuming the opposite,

因為每個盒子都要有 2 顆以上的紅寶石。

that two boxes have 7 or fewer.

在這個情況下，第三個盒子

Those could not be the two that differ by 6,

會有至少 13 顆紅寶石——即 7+6。

because every box must have at least 2 rubies.

把三個盒子相加起來，

In that case, the third box would have at most 13 rubies—that's 7 plus 6.

最大的總合是 27。

Add up all three of those boxes,

那還不到 30 顆，

and the most that could equal is 27.

這個情境就是不可能的。

Since that's less than 30,

現在，用所謂的反證法，你能知道

this scenario isn't possible.

其中兩個盒子有至少 8 顆紅寶石。

You now know, by what's called a proof by contradiction,

如果你對每個盒子都索取 8 顆，

that two of the boxes have 8 or more rubies.

你就能得到至少 16 顆——

If you ask for 8 from all three boxes

你能看出 這是你能做的最好保證，

you'll receive at least 16—

若你能重新再想一次 8 顆、14 顆、8 顆的情境的話。

and that's the best you can guarantee,

你取回了這個惡棍 一半以上的財富，

as you can see by thinking again about the 8, 14, 8 scenario.

做為給大眾的賠償。

You've recovered more than half the scoundrel's fortune

雖然他還是得以保住一些紅寶石，

as restitution for the public.

他的財富肯定是失去了一些光芒。

And though he's managed to hold on to some of his rubies,

his fortune has definitely lost some of its shine.