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