Can you solve the demon dance party riddle? - Edwin Meyer

譯者: Veronica Wu 審譯者: Helen Chang

Once each year, thousands of logicians descend into the desert for Learning Man,

每年，上千個邏輯學家

會為了參與為期一週的 「 Learning Man 」活動進入沙漠，

a week-long event they attend to share their ideas,

來分享彼此的想法、

think through tough problems... and mostly to party.

深思困難的問題...... 並以交際聚會為主。

And at the center of that gathering is the world’s most exclusive club,

在人群中間的， 是世界上最嚴選的俱樂部。

where under the full moon, the annual logician’s rave takes place.

一年一度邏輯學家的狂歡， 在滿月下舉行。

The entry is guarded by the Demon of Reason,

入口是由理性魔鬼守衛的，

and the only way to get in is to solve one of his dastardly challenges.

且唯一能夠進入的方式 就是解開他提出的卑鄙問題。

You’re attending with 23 of your closest logician friends,

你是與其他 23 個最要好的 邏輯學家朋友一起參加的，

but you got lost on the way to the rave and arrived late.

但你因為在途中迷路而晚到。

They're already inside, so you must face down the demon alone.

他們已經在裡面了， 所以你必須獨自面對魔鬼。

He poses you the following question:

他提出以下問題：

When your friends arrived, the demon put masks on their faces

當你的朋友們到達時， 魔鬼讓他們戴上面具

and forbade them from communicating in any way.

並禁止他們用任何方式溝通。

No one at any point could see their own masks,

沒有人可以看到自己的面具，

but they stood in a circle where they could see everyone else’s.

但他們會圍成一個圈 使大家可以看到彼此的。

The demon told the logicians that he distributed the masks in such a way

魔鬼告訴邏輯學家們 他分配面具的方式

that each person would eventually be able to figure out their mask’s color

讓每個人最終都可以光靠邏輯推理

using logic alone.

來判斷自己的面具顏色。

Then, once every two minutes, he rang a bell.

接者，他每兩分鐘響一次鈴。

At that point, anyone who could come to him

鈴響時，任何人都可以到他那裡

and tell him the color of their mask would be admitted.

並告訴他認為自己的 面具顏色為何，即可通關。

Here’s what happened:

事情是這樣發展的：

Four logicians got in at the first bell.

四個邏輯學家 在第一次鈴響時進入。

Some number of logicians, all in red masks, got in at the second bell.

若干個紅色面具的邏輯學家， 在第二次鈴響時進入。

Nobody got in when the third bell rang.

第三次鈴響時沒有人進入。

Logicians wearing at least two different colors got in at the fourth bell.

有至少兩種面具顏色的邏輯學家 在第四次鈴響時進入。

All 23 of your friends played the game perfectly logically

你所有 23 位朋友 都用完美的邏輯在玩這個遊戲

and eventually got inside.

且最終都進入了會場。

Your challenge, the demon explains,

魔鬼說，你的挑戰，

is to tell him how many people gained entry when the fifth bell rang.

就是告訴他在第五次響鈴時 有多少人進入會場。

Can you get into the rave?

你能進去狂歡嗎？

Pause here to figure it out yourself.

在這裏按下暫停 以自行作答

Answer in 3

[ 在 3 秒內回答 ]

Answer in 2

[ 在 2 秒內回答 ]

Answer in 1

[ 在 1 秒內回答 ]

It’s initially difficult to imagine how anyone could,

一開始很難想像，

為何有人可以光靠邏輯推理 和其他人的面具顏色

using just logic and the colors they see on the other masks,

deduce their own mask color.

來推斷出自己的面具顏色。

But even before the first bell, everyone will realize something critical.

但甚至在第一個鈴響之前， 每個人都會發現一些重要的事。

Let’s imagine a single logician with a silver mask.

讓我們想像一下 只有一個邏輯學家戴銀色面具。

When she looks around, she’d see multiple colors, but no silver.

當她環顧周圍，會看到 各種顏色，但沒有銀色。

So she couldn’t ever know that silver is an option,

所以她永遠不會知道 銀色是一種可能，

making it impossible for her to logically deduce that she must be silver.

因此她不可能用邏輯推斷出 她的面具是銀色。

That contradicts rule five, so there must be at least two masks of each color.

那就與第五項規則矛盾了， 所以每種顏色的面具至少有兩個。

Now, let’s think about what happens

現在，來想像一下

當剛好兩個人戴 相同顏色的面具時會發生什麼事。

when there are exactly two people wearing the same color mask.

Each of them sees only one mask of that color.

他們兩個各會看到 有一個顏色的面具只有一個人戴。

But because they already know that it can’t be the only one,

但因為他們已經知道 那不可能是唯一的一個，

they immediately know that their own mask is the other.

他們馬上就知道 自己是另外一個。

This must be what happened before the first bell:

這勢必是第一聲鈴響前 所發生的事：

two pairs of logicians each realized their own mask colors

兩對邏輯學家 在看見空間裡唯一的顏色時，

when they saw a unique color in the room.

得知了自己面具顏色。

What happens if there are three people wearing the same color?

如果有三個人戴著 相同顏色的面具會發生什麼事？

Each of them—A, B and C— sees two people with that color.

A 、 B 和 C 各會看到 兩個人戴著那個顏色。

From A’s perspective, B and C would be expected to behave the same way

從 A 的角度來看，他認為 B 和 C 會與橘色和紫色的成對面具一樣

that the orange and purple pairs did, leaving at the first bell.

在第一聲鈴響時離開。

When that doesn’t happen,

但當這件事沒發生時，

each of the three realizes that they are the third person with that color,

他們三個就會知道 他們是戴著那個顏色的第三個人，

and all three leave at the next bell.

而在下一聲鈴響時離開。

That was what the people with red masks did—

那是戴著紅色面具的人所做的事——

so there must have been three of them.

所以勢必有三個人戴著紅色面具。

We’ve now established a basis for inductive reasoning.

我們現在已經建立了歸納推理的基礎。

Induction is where we can solve the simplest case,

歸納法是去解決最小問題，

then find a pattern that will allow the same reasoning

然後找出同樣的推理模式

to apply to successively larger sets.

來依序應用在更大的問題當中。

The pattern here is that everyone will know what group they’re in

這裡的模式是

只要較小的顏色群組有機會離開， 每個人就會知道自己屬於哪一群。

as soon as the previously sized group has the opportunity to leave.

After the second bell, there were 16 people.

在第二聲鈴響後， 會剩下 16 個人。

No one left on the third bell,

在第三聲鈴響時沒有人離開，

so everyone then knew there weren’t any groups of four.

所以大家知道 沒有任何四人的群組。

Multiple groups, which must have been of five,

有不只一個五人以上的群組，

left on the fourth bell.

在第五聲鈴響時離開。

Three groups would leave a solitary mask wearer,

若有三個組離開 會留下一個單獨的人，

which isn’t possible, so it must’ve been two groups.

那是不可能的， 所以勢必是兩組。

And that leaves six logicians outside when the fifth bell rings:

然後在第五聲鈴響時 剩下邏輯學家六個還在外面：

the answer to the demon’s riddle.

也就是魔鬼要的解答。

Nothing left to do but join your friends and dance.

剩下的事情只有 和你的朋友們跳舞了。