who needs to declare his successor.

你是一位古怪國王 身邊的首席顧問，

He wants his heir to be good at arithmetic, lucky,

國王需要宣佈他的繼承人。

and above all else, honest.

他希望繼承人的特質 要包括擅長算術、運氣好，

So he's devised a competition to test his children,

以及最重要的，誠實。

and ordered you to choose the winner.

所以他設計了一項競賽 來測試他的孩子們，

Each potential heir will be given the same two six-sided dice.

並命令你選出贏家。

The red die has the numbers 2, 7, 7, 12, 12, and 17.

每位可能的繼承人都會拿到 同樣的兩顆六面骰子。

The blue one has 3, 8, 8, 13, 13, and 18.

紅骰子上的數字是 2、7、

The dice are fair, so each side is equally likely to come up.

7、12、12、17。

Each contestant will be sent into a Royal Rolling Room,

藍骰子上的數字是 3、8、

where they'll roll both dice 20 times.

8、13、13、18。

A contestant's score starts at zero, and each turn,

骰子是公平的，每一面 向上的機率都一樣。

they should add the total of the two numbers rolled to their score.

每位參賽者都會被 送入一間皇家擲骰室，

After 20 turns, they should report their final score.

在那裡，他們要將 兩顆骰子擲 20 次。

The rooms are secure, and no one observes the rolls.

參賽者一開始的分數 是 0，每一輪之後，

That means a contestant could add incorrectly, or worse, be dishonest

他們要把兩個擲出的數字總和 加到他們的分數上。

and make up a score they didn't achieve.

20 輪之後，他們要 回報他們的最終分數。

This is where you come in.

這些房室很安全， 沒人能看到擲出的結果。

The king has instructed you that if you're at least 90% sure a contestant

那就表示，參賽者可能會加錯，

mis-added or cheated, you should disqualify them.

更糟的是，還可能說謊，

The highest-scoring player who remains will be the new heir to the throne.

掰出一個他們沒有達到的分數。

After you explain the rules, the children run to their rooms.

這就是你派上用場的時候了。

When they return, Alexa announces her score is 385.

國王指示你，如果你有 90% 以上的把握

Bertram says 840. Cassandra reports 700. And Draco declares 423.

認為某位參賽者加錯 或作弊，就要將他除名。

The future of the kingdom is in your hands.

得分最高的參賽者 就會是王位的新繼承人。

Whom do you proclaim to be the worthiest successor?

你解釋完規則之後，這些 孩子們衝進他們的房室中。

Pause now to figure it out for yourself.

他們回來時，艾麗莎 宣佈她的分數為 385。

Upon inspection, most of these scores are concerning.

柏特姆說 840。

Let's start with the highest.

卡珊卓回報 700。

Bertram scored 840.

卓克聲稱 423。

That's impressive… but is it even possible?

王國的未來操在你的手中。

The highest numbers on the two dice are 17 and 18.

你會宣佈誰是最配得上的繼承人？

17 plus 18 is 35, so in 20 rolls,

如果你想自行解題，請在此暫停。

the greatest possible total is 20 times 35, or 700.

檢閱之後覺得 大部分的分數令人關切。

Even if Bertram rolled all the highest numbers,

咱們從最高分開始。

he couldn't have scored 840.

柏特姆得到 840 分。

So he's disqualified.

令人印象深刻……但有可能嗎？

Cassandra, the next-highest roller, reported 700.

兩顆骰子上最大的數字 是 17 和 18。

That's theoretically possible… but how hard is it to be that lucky?

17 加上 18 等於 35， 所以若擲 20 次，

In order to get 700,

可能得到的最高分數 是 20 乘以 35，

Cassandra would have to roll the highest number out of six

即 700。

on 40 separate occasions.

就算柏特姆擲出所有的最大數字，

The probability of this is 1 over 6 to the 40th power,

他也不可能得到 840 分。

or 1 in about 13 nonillion— that's 13 followed by 30 zeros.

所以他被除名了。

To put that in perspective, there are about 7.5 billion people in the world,

分數次高的是卡珊卓， 她回報 700 分。

and 7.5 billion squared is a lot less than 13 nonillion.

理論上是可能的……

Rolling the highest number all 40 times is much less likely

但要那麼幸運有多困難？

than if you picked a completely random person on Earth,

若要得到 700 分，

and it turned out to be actor Paul Rudd…

卡珊卓必須要在 40 個 分別的場合中

and then you randomly picked again, and got Paul Rudd again!

都擲出六個數字中最大的。

You can't be 100% sure that Cassandra's score didn't happen by chance…

機率是 1 除以 6 的 40 次方，

but you can certainly be 90% sure, so she should be disqualified.

大約是 13 乘以 10 的 30 次方分之 1，

Next up is Draco, with 423.

也就是分母是 13 後面 再加 30 個 0。

This score isn't high enough to be suspicious.

更清楚地說， 全世界大約有 75 億人，

But it's impossible for a different reason.

而 75 億的平方還是比 13 乘以 10 的 30 次方小很多。

Pick a number from each die, and add them up.

40 次全擲出最大數字的機率遠小於

No matter which combination you choose, the result ends in a 0 or a 5.

你在地球上隨機挑選一個人

That's because every red number is 2 more than a multiple of 5,

結果發現你挑到了 演員保羅 · 路德……

and every blue number is 3 more than a multiple of 5.

接著你再次隨機挑選， 又再挑到保羅 · 路德的機率！

This means that when you add them together,

你不能 100% 肯定 卡珊卓的分數不是巧合……

you'll always get an exact multiple of 5.

但你有九成的把握，

And when you add rolls that are multiples of 5,

所以她也應該被除名。

the result will also be a multiple of 5.

下一位是卓克，423 分。

These sorts of relationships between integers are studied

這個分數沒有高到可疑。

in a branch of math called number theory.

但有另一個原因說明 它是不可能達到的。

Here number theory shows us that Draco's score,

從每個骰子上挑 一個數字，通通加起來。

which is not a multiple of 5, cannot be achieved.

不論你選到什麼樣的組合，

So he should be disqualified as well.

總分的尾數都會是 0 或 5。

This leaves Alexa, whose score is a multiple of 5

那是因為每個紅色數字 都是 5 的倍數加 2，

and is in the achievable range.

每個藍色數字都是 5 的倍數加 3。

In fact, the most likely score is 400, so she was a little bit unlucky.

那就表示當你把兩個數字加總時，

But with everyone else disqualified, she's the last heir standing.

一定會得到 5 的倍數。

All hail Queen Alexa, the worthiest successor!

每一輪的總分都是 5 的倍數，

At least if you agree that the best way to organize your government

通通加總也仍然會是 5 的倍數。

is a roll of the dice...

有一個數學分支叫做「數論」