Can you solve the stolen rubies riddle? - Dennis Shasha

3,723,869 views ・ 2018-10-25

TED-Ed


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譯者: Lilian Chiu 審譯者: Ann Chen
00:07
One of the kingdom’s most prosperous merchants
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王國當中最富有的商人之一
00:09
has been exposed for his corrupt dealings.
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過去骯髒的交易被爆料出來。
00:13
Nearly all of his riches
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他幾乎把所有的財富都投資到
00:14
are invested in a collection
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00:16
of 30 exquisite Burmese rubies,
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一系列共三十顆精緻 緬甸紅寶石的收藏品上,
00:19
and the crowd in the square
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而廣場上的群眾喧鬧著
00:21
is clamoring for their confiscation
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要將這些紅寶石沒收 來償還他欠受害者的錢。
00:23
to reimburse his victims.
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00:25
But the scoundrel and his allies at court
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但這個惡棍和他在法庭上的盟友
00:28
have made a convincing case
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提出了很有說服力的論據,
00:29
that at least some of his wealth
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說明他的財富至少 有一部分是合法取得,
00:31
was obtained legitimately,
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00:33
and through good service to the crown.
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並且是為君王效力而得。
00:35
The king ponders for a minute
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國王沉思了一會兒, 宣佈了他的判決。
00:37
and announces his judgment.
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00:39
Because there’s no way to know
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因為不可能知道 有多少比例的紅寶石
00:40
which portion of the rubies were bought with ill-gotten wealth,
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是用非法取得之財富所購買,
00:44
the fine will be determined through a game of wits between the merchant
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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。
00:48
and the king’s most clever advisor – you.
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00:52
You’re both told the rules in advance.
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兩位在賽前都已得知遊戲規則。
00:54
The merchant will be allowed to discreetly divide his rubies
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商人被允許可以私底下 將他的紅寶石
00:57
among three boxes, which will then be placed in front of you.
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分別放入三個盒子中, 接著盒子會被放在你面前。
01:02
You will be given three cards,
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你會拿到三張卡片,
01:04
and must write a number between 1 and 30 on each,
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你必須要在每張卡片上寫下 1 到 30 之間的一個數字,
01:08
before putting a card in front of each of the boxes.
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接著在每個盒子前面 各放一張卡片。
01:12
The boxes will then all be opened.
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再來盒子會被打開。
01:15
For each box, you will receive exactly
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你能從每個盒子得到的紅寶石數目
01:17
as many rubies as the number written on the corresponding card,
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就是你寫在對應卡片上的數字。
01:21
if the box has that many.
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前提是那個盒子中的 紅寶石要足夠。
01:23
But if your number is greater than the number of rubies actually there,
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但如果你的數字比盒子中 實際的紅寶石數目還多,
01:27
the scoundrel gets to keep the entire box.
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這個惡棍就可以保住整個盒子。
01:30
The king puts just two constraints on how the scoundrel distributes his rubies.
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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。
01:36
Each box must contain at least two rubies
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每個盒子要有 2 顆以上的紅寶石,
01:40
and one of the boxes must contain exactly six more rubies than another—
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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——
01:45
but you won’t know which boxes those are.
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但你不知道是哪個盒子。
01:48
After a few minutes of deliberation,
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在深思熟慮了幾分鐘之後,
01:50
the merchant hides the gems, and the boxes are brought in front of you.
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商人把寶石藏起來, 接著盒子被送到你面前。
01:54
Which numbers should you choose
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你要選擇哪些數字
01:56
in order to guarantee the largest possible fine for the scoundrel
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才能確保讓這個惡棍 被罰最高的罰金,
02:00
and the greatest compensation for his victims?
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讓他的受害者得到最高額的補償?
02:03
Pause the video now if you want to figure it out for yourself.
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若你想要嘗試自己解題, 請在這裡暫停。
02:07
Answer in 3
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答案即將公佈:3
02:08
Answer in 2
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答案即將公佈:2
02:09
Answer in 1
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答案即將公佈:1
02:11
You don’t want to overshoot by being too greedy.
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你不會想要因為太貪心 而讓數字超過。
02:14
But there is a way you can guarantee
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但有一個方法,可以確保
02:16
to get more than half of the scoundrel’s stash.
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這個惡棍必須要交出 他至少一半的藏匿物。
02:20
The situation resembles an adversarial game like chess –
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這個情況就類似 對抗遊戲,如西洋棋——
02:24
only here you can’t see the opponent’s position.
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只是在這裡, 你看不到對手的位置。
02:27
To figure out the minimum number of rubies you’re guaranteed to win,
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若要得知你最少一定能 得到的紅寶石數目,
02:31
you need to look for the worst case scenario,
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你得要去探究最糟的情境,
02:34
as if the merchant already knew your move
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也就是當作商人 已經知道你要怎麼做,
02:36
and could arrange the rubies to minimize your winnings.
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他能夠安排紅寶石的分配, 來讓你贏得最少。
02:40
Because you have no way of knowing which boxes will have more or fewer rubies,
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因為你不可能知道哪個盒子 有比較多/少紅寶石,
02:44
you should pick the same number for each.
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你應該對所有盒子 都選一樣的數字。
02:47
Suppose you write three 9’s.
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假設你寫了三個 9。
02:49
The scoundrel might have allocated the rubies as 8, 14 and 8.
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這個惡棍可以把紅寶石 分配成 8 顆、14 顆、8 顆。
02:54
In that case, you’d receive 9 from the middle box and no others.
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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。
02:59
On the other hand, you can be sure that at least two boxes
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另一方面,你能確定 至少有兩個盒子
03:02
have a minimum of 8 rubies.
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裡面有至少 8 顆紅寶石。
03:04
Here’s why.
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原因如下。
03:05
We’ll start by assuming the opposite,
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我們先從相反的假設開始,
03:08
that two boxes have 7 or fewer.
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兩個盒子中的紅寶石 數目只有 7 顆或以下。
03:10
Those could not be the two that differ by 6,
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這兩個盒子的寶石數目 不可能相差 6 顆,
03:13
because every box must have at least 2 rubies.
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因為每個盒子都要有 2 顆以上的紅寶石。
03:16
In that case, the third box would have at most 13 rubies—that’s 7 plus 6.
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在這個情況下,第三個盒子
會有至少 13 顆紅寶石——即 7+6。
03:23
Add up all three of those boxes,
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把三個盒子相加起來,
03:25
and the most that could equal is 27.
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最大的總合是 27。
03:28
Since that’s less than 30,
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那還不到 30 顆,
03:30
this scenario isn’t possible.
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這個情境就是不可能的。
03:32
You now know, by what’s called a proof by contradiction,
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現在,用所謂的反證法,你能知道
03:35
that two of the boxes have 8 or more rubies.
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其中兩個盒子有至少 8 顆紅寶石。
03:39
If you ask for 8 from all three boxes
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如果你對每個盒子都索取 8 顆,
03:42
you’ll receive at least 16—
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你就能得到至少 16 顆——
03:44
and that’s the best you can guarantee,
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你能看出 這是你能做的最好保證,
03:46
as you can see by thinking again about the 8, 14, 8 scenario.
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若你能重新再想一次 8 顆、14 顆、8 顆的情境的話。
03:51
You’ve recovered more than half the scoundrel’s fortune
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你取回了這個惡棍 一半以上的財富,
03:54
as restitution for the public.
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做為給大眾的賠償。
03:56
And though he’s managed to hold on to some of his rubies,
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雖然他還是得以保住一些紅寶石,
03:59
his fortune has definitely lost some of its shine.
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他的財富肯定是失去了一些光芒。
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