Can you solve the stolen rubies riddle? - Dennis Shasha

譯者: Lilian Chiu 審譯者: Ann Chen

One of the kingdom’s most prosperous merchants

王國當中最富有的商人之一

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

of 30 exquisite Burmese rubies,

一系列共三十顆精緻 緬甸紅寶石的收藏品上，

and the crowd in the square

而廣場上的群眾喧鬧著

is clamoring for their confiscation

要將這些紅寶石沒收 來償還他欠受害者的錢。

to reimburse his victims.

But the scoundrel and his allies at court

但這個惡棍和他在法庭上的盟友

have made a convincing case

提出了很有說服力的論據，

that at least some of his wealth

說明他的財富至少 有一部分是合法取得，

was obtained legitimately,

and through good service to the crown.

並且是為君王效力而得。

The king ponders for a minute

國王沉思了一會兒， 宣佈了他的判決。

and announces his judgment.

Because there’s no way to know

因為不可能知道 有多少比例的紅寶石

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

是用非法取得之財富所購買，

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

罰金將要透過 一個益智比賽來決定，

比賽是由這名商人對抗 國王最聰明的顧問——你。

and the king’s most clever advisor – you.

You’re both told the rules in advance.

兩位在賽前都已得知遊戲規則。

The merchant will be allowed to discreetly divide his rubies

商人被允許可以私底下 將他的紅寶石

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

分別放入三個盒子中， 接著盒子會被放在你面前。

You will be given three cards,

你會拿到三張卡片，

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

你必須要在每張卡片上寫下 1 到 30 之間的一個數字，

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

接著在每個盒子前面 各放一張卡片。

The boxes will then all be opened.

再來盒子會被打開。

For each box, you will receive exactly

你能從每個盒子得到的紅寶石數目

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

就是你寫在對應卡片上的數字。

if the box has that many.

前提是那個盒子中的 紅寶石要足夠。

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

但如果你的數字比盒子中 實際的紅寶石數目還多，

the scoundrel gets to keep the entire box.

這個惡棍就可以保住整個盒子。

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

國王只設定了兩個條件來限制

惡棍要如何分配他的紅寶石。

Each box must contain at least two rubies

每個盒子要有 2 顆以上的紅寶石，

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

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——

but you won’t know which boxes those are.

但你不知道是哪個盒子。

After a few minutes of deliberation,

在深思熟慮了幾分鐘之後，

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

商人把寶石藏起來， 接著盒子被送到你面前。

Which numbers should you choose

你要選擇哪些數字

in order to guarantee the largest possible fine for the scoundrel

才能確保讓這個惡棍 被罰最高的罰金，

and the greatest compensation for his victims?

讓他的受害者得到最高額的補償？

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

若你想要嘗試自己解題， 請在這裡暫停。

Answer in 3

答案即將公佈：3

Answer in 2

答案即將公佈：2

Answer in 1

答案即將公佈：1

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

你不會想要因為太貪心 而讓數字超過。

But there is a way you can guarantee

但有一個方法，可以確保

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

這個惡棍必須要交出 他至少一半的藏匿物。

The situation resembles an adversarial game like chess –

這個情況就類似 對抗遊戲，如西洋棋——

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

只是在這裡， 你看不到對手的位置。

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

若要得知你最少一定能 得到的紅寶石數目，

you need to look for the worst case scenario,

你得要去探究最糟的情境，

as if the merchant already knew your move

也就是當作商人 已經知道你要怎麼做，

and could arrange the rubies to minimize your winnings.

他能夠安排紅寶石的分配， 來讓你贏得最少。

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

因為你不可能知道哪個盒子 有比較多／少紅寶石，

you should pick the same number for each.

你應該對所有盒子 都選一樣的數字。

Suppose you write three 9’s.

假設你寫了三個 9。

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

這個惡棍可以把紅寶石 分配成 8 顆、14 顆、8 顆。

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

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。

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

另一方面，你能確定 至少有兩個盒子

have a minimum of 8 rubies.

裡面有至少 8 顆紅寶石。

Here’s why.

原因如下。

We’ll start by assuming the opposite,

我們先從相反的假設開始，

that two boxes have 7 or fewer.

兩個盒子中的紅寶石 數目只有 7 顆或以下。

Those could not be the two that differ by 6,

這兩個盒子的寶石數目 不可能相差 6 顆，

because every box must have at least 2 rubies.

因為每個盒子都要有 2 顆以上的紅寶石。

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

在這個情況下，第三個盒子

會有至少 13 顆紅寶石——即 7+6。

Add up all three of those boxes,

把三個盒子相加起來，

and the most that could equal is 27.

最大的總合是 27。

Since that’s less than 30,

那還不到 30 顆，

this scenario isn’t possible.

這個情境就是不可能的。

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

現在，用所謂的反證法，你能知道

that two of the boxes have 8 or more rubies.

其中兩個盒子有至少 8 顆紅寶石。

If you ask for 8 from all three boxes

如果你對每個盒子都索取 8 顆，

you’ll receive at least 16—

你就能得到至少 16 顆——

and that’s the best you can guarantee,

你能看出 這是你能做的最好保證，

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

若你能重新再想一次 8 顆、14 顆、8 顆的情境的話。

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

你取回了這個惡棍 一半以上的財富，

as restitution for the public.

做為給大眾的賠償。

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

雖然他還是得以保住一些紅寶石，

his fortune has definitely lost some of its shine.

他的財富肯定是失去了一些光芒。