Can you solve the cheating royal riddle? - Dan Katz

翻译人员: Siyu Yang 校对人员: Zizhuo Liang

You’re the chief advisor to an eccentric king

你是一个古怪国王的首席顾问，

who needs to declare his successor.

国王需要宣布他的继承人。

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

他希望他的继承人 要擅长算数，运气好，

and above all else, honest.

以及最重要的是，要诚实。

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

所以他设计了一个比赛 来考验他的孩子们，

and ordered you to choose the winner.

并命令你来选出获胜者。

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

每个可能的继承人都会 拿到两枚相同的六面骰子。

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

红色的骰子上的数字是 2，7，7，12，12 和 17 。

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

蓝色骰子上的数字是 3，8，8，13, 13 和 18。

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

骰子是公平的，这意味着 每面向上的概率都相同。

Each contestant will be sent into a Royal Rolling Room,

每位参赛者都会被 送去一个皇家掷骰室，

where they’ll roll both dice 20 times.

在那他们将两个骰子各掷 20 次。

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

在每回合开始时 参赛者们的分数都是零。

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

每轮结束后他们要把两个数字的 总和加到他们的分数中。

After 20 turns, they should report their final score.

20 个回合后，他们要上报 他们的最后得分。

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

这些房间很安全， 没人有能看到掷骰的过程。

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

这就意味着参赛者可能会加错， 或更糟的是，可能会撒谎

and make up a score they didn’t achieve.

并编造出一个他们没有达到的分数。

This is where you come in.

这就是你要履行职责的时刻了。

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

国王下令如果你有 90% 的 把握确定有某个选手

mis-added or cheated, you should disqualify them.

添加错误或舞弊， 你就应该取消他的资格。

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

得分最高的选手 将成为新的王位继承人。

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

当你解释完规则后， 孩子们跑进他们的房间。

When they return, Alexa announces her score is 385.

当他们回来时，

艾丽莎宣布她的成绩是 385，

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

伯特伦说 840，

卡桑德拉说 700，

德拉科说 423。

The future of the kingdom is in your hands.

王国的未来掌握在你手上。

Whom do you proclaim to be the worthiest successor?

你将宣布谁是最合适的继承人？

Pause now to figure it out for yourself.

【如果你想尝试自行解题，请在此暂停。】

Upon inspection, most of these scores are concerning.

经检查，大部分的分数令人起疑。

Let’s start with the highest.

我们先从最高分开始。

Bertram scored 840.

伯特伦得了 840 分。

That’s impressive… but is it even possible?

这真了不起……但是这可能吗？

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

两个骰子上最大的数字是 17 和 18 。

17 plus 18 is 35, so in 20 rolls,

17 加 18 是 35，所以在 20 个回合中，

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

可能得到的最高分是 20 乘以 35 ，即 700。

Even if Bertram rolled all the highest numbers,

就算伯特伦掷出所有最大的数字，

he couldn’t have scored 840.

他也不可能得到 840 分。

So he’s disqualified.

所以他的资格被取消了。

Cassandra, the next-highest roller, reported 700.

得分第二的掷骰者是卡桑德拉，

她取得了 700 分。

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

这在理论上是可能的…… 但是做到这样幸运要多难呢？

In order to get 700,

如果想要取得 700 分，

Cassandra would have to roll the highest number out of six

卡桑德拉必须要在 40 种不同情况下

on 40 separate occasions.

掷出六个数字中最大的数。

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

这个概率是六分之一的 40 次方。

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

或者说是 13 乘以 10 的 30 次方分之一，

也就是 13 后面有 30 个 0 。

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

换个角度来看，地球上约有 75 亿人，

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

75 亿的平方还是比 13 乘以 10 的 30 次方要小得多，

Rolling the highest number all 40 times is much less likely

在全部 40 次投掷中都掷出最大的数

than if you picked a completely random person on Earth,

比在地球上随机挑选一个人，

而那个人正好是演员 保罗 · 路德（Paul Rudd）

and it turned out to be actor Paul Rudd…

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

接着再随机挑选又挑到保罗 · 路德的

概率还要小得多。

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

你不能百分百肯定卡桑德拉的

分数不是巧合……

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

但是你至少有九成把握她作弊了， 所以她应该被取消资格。

Next up is Draco, with 423.

下一位是德拉科，423 分。

This score isn’t high enough to be suspicious.

这个分数并没有高到受怀疑的程度。

But it’s impossible for a different reason.

但有另一个原因， 可以说明这是不可能的。

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

从每个骰子中选一个数字， 然后把它们加起来。

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

无论你选择哪种组合， 结果都应该以 0 或 5 结尾。

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

这是因为每个红色的数字 都比 5 的倍数大 2 ，

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

每个蓝色的数字都比 5 的倍数大 3，

This means that when you add them together,

这就意味着当你把它们加在一起时，

you’ll always get an exact multiple of 5.

你总会得到 5 的倍数。

And when you add rolls that are multiples of 5,

并当你把两个 5 的倍数相加时，

the result will also be a multiple of 5.

结果一定是 5 的倍数。

These sorts of relationships between integers are studied

有一个数学分支叫做 “数论”

就是在研究整数之间的关系。

in a branch of math called number theory.

Here number theory shows us that Draco’s score,

在这里“数论”告诉我们，

德拉科的分数不是 5 的倍数， 因此是无法成立的。

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

So he should be disqualified as well.

所以他也应该被取消资格。

This leaves Alexa, whose score is a multiple of 5

现在只剩下艾丽莎， 她的分数是 5 的倍数，

and is in the achievable range.

并在一个可实现的范围内。

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

事实上，最有可能的分数是 400， 所以她并不算走运。

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

但因为其他人都被取消了资格， 她成为了唯一的继承人。

All hail Queen Alexa, the worthiest successor!

向艾丽莎女王致敬， 她是最配得上的继承人！

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

当然，这前提是你认同 组建政府最好的方式是掷骰子……

is a roll of the dice...