This one weird trick will get you infinite gold - Dan Finkel

譯者: 子菱 方 審譯者: C Leung

Well, you're in a real pickle.

唉！你真的遇到了大麻煩。

You see, a few years ago, the king decided your life would be forfeit

國王多年前決定取你性命，

unless you tripled the gold coins in his treasury.

除非你有辦法使他 寶庫裡的金幣變成三倍。

Fortunately for you, a strange little man appeared

幸運的是，

一個奇怪的小矮人出現在你面前

and magically performed the feat.

並神奇地完成壯舉。

He placed handfuls of coins in and out of a magical bag,

他將滿手的金幣 從神奇的袋子不斷拿進拿出，

and sang a strange rhyme:

並哼唱著奇怪的曲調：

“The more gold goes in and more comes out, as sure as I am me.

「 進去的黃金越多， 出來的就越多，我非常肯定。」

And in again, and out again, and now it’s it times three!”

再放進去，再拿出來 ， 現在變成三倍了！

Incredibly, that tripled the coins and saved your life.

難以置信地， 金幣變了三倍並拯救了你的命。

Were you grateful? Yes. Were you desperate? Yes.

你感激嗎？ 非常。

你絕望嗎？ 對。

Did you promise him your first-born child in exchange for his help? Yes.

你曾向他承諾以你第一個 出生的孩子換取他的幫助嗎？

是的。

Fast forward to today.

很快回到今天，

No sooner have you given birth to a beautiful baby boy

當你剛生下一個可愛的男寶寶，

than the little man shows up to claim his prize.

小矮人就出現，想領取他的獎勵。

You cry and beg him not to take the baby.

你大哭並乞求他不要帶走寶寶。

Softening, he begins, “If you can guess my name—”

他緩緩地開口： 「如果你能猜到我的名字—」

“Banach-Tarski?” you say. “It’s on the front of your shirt.”

「巴拿赫...塔斯基?」你回答。 「就寫在你的衣服上。」

“What! That won’t do. Aha. My bag,” he explains,

「什麼！那不算數。 啊，我的袋子...」他接著說明：

“increases the number of gold coins placed inside it in a very special way.

「增加放在裡面的金幣， 其實是用一個很特別的方法，

If I take any number of coins and place them in, more will come out.

我如果拿任一數量的金幣放在裡面 ， 將會有更多的金幣出來。

And if I place those in the bag again,

如果我再次把它們放進袋子裡，

the total that comes out will be three times whatever I began with.”

不論開始的數量， 得到的總數將會變成三倍 。」

He takes 13 coins and places them in the bag,

他拿出 13 枚金幣並放進袋子，

then removes the contents.

然後再取出來。

“I’ve used the magic once, not twice,” he says.

「我已使用了一次魔法， 並非兩次。」他說。

“Tell me how many coins are in my hand and I’ll have mercy.”

「告訴我我手裡有多少金幣， 我就饒了你。」

How many coins is he holding?

他手裡有多少金幣呢？

Pause here to figure it out yourself. Answer in 3

請在此暫停並想想看 。 倒數後公布，三

Answer in 2

二

Answer in 1

一

The bag’s magic works just like what in mathematics is called a “function,”

袋子的魔法其實就像 數學裡的「函數」，

and it’s convenient in both cases to use an arrow to denote the transformation.

而在兩個例子中， 可以很方便的用箭頭說明過程。

We can write what we know like this.

我們可以把已知的寫成這樣。

We want to know what goes in this particular blank.

我們想知道這指定的空格。

Maybe the bag just multiplies the number of coins by some number.

或許這袋子只將金幣數量乘以某個數字。

In that case, multiplying by that number twice

這樣的話 ，將數字相乘兩次

would be the same as multiplying by 3,

將等於乘上 3 的值，

which means the multiplier would be the square root of 3.

這意味著乘數將是 3 的平方根。

That’s not a whole number, though.

但這並非整數，

And we don’t have bits of gold coins coming out of the bag.

而且我們沒有金幣碎能從袋子裡拿出來。

Something else is going on.

還有別的可能性。

Well, if filling in the blank between 13 and 39 is too hard,

如果填入介於 13 到 39 的數字太難了，

maybe we can start with something easier.

或許我們可以從較簡單的開始。

Can we figure out what’ll happen to 1 coin?

我們可以知道 1 枚金幣會如何嗎？

If you use the bag on a single coin twice, you end up with triple;

如果你用 1 枚金幣放袋子兩次， 你最後會得到 3 倍，

that’s three gold pieces.

正是 3 枚金幣。

Because the bag always increases the number of gold coins,

由於袋子永遠增加金幣數量，

the blank must be between 1 and 3, so 2.

空格處必介於 1 和 3，所以是 2 。

It’s a start. What’s next?

這只是一個開始，接下來呢？

Let’s think about a few other possible starting places.

我們再想想其他可能起始點。

We already know 2 becomes 3

我們已知 2 變 3，

and that lets us fill in the next blank as well.

然後會讓我們填入其餘的空格。

Now we’re getting somewhere!

現在我們已經找到脈絡！

We just need to extend this out to 13.

我們只需把它延伸到 13。

Remember the other rule, though:

但請記得另外一個規則：

when you put more coins in, you get more coins out.

當你放入越多金幣，你將拿到更多金幣。

That means the numbers in every column must go in increasing order as well.

這表示在欄位的數字 也必須呈遞增的序列。

In other words, because 6 coins become 9,

也就是說， 由於 6 枚金幣變成 9，

it’s not possible for 4 coins to become 10.

4 枚金幣就不可能變成 10。

Nor could 4 become 5, since 3 becomes 6.

4 也不可能變成 5， 因為 3 變成 6。

So 7 and 8 fill those blanks on the right of 4 and 5,

所以將 7 和 8 分別 填入 4 和 5 右方的空格，

which in turn gives the answer for two more blanks.

我們可以得到另外兩個空格的答案。

Knowing that the numbers go in increasing order in every column,

已知數字在每一欄位呈遞增的序列，

the only choices for the remaining blanks are 19, 20, 22, and 23.

剩下空格的唯一選擇 只有19、20、22 和 23。

And look! We have our answer!

看吧！我們得到答案了！

There must be 22 gold coins in his hand.

他手中一定有 22 枚金幣。

“I’ll give you three guesses,” the little man begins to say.

「我會給你三次猜題機會，」 小矮人開始說道。

“22 coins,” you respond.

「22 枚金幣，」 你回答。

“What?! How did you know?”

「什麼？！你怎知道的？」

“I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”

「我很喜歡這個很棒的謎題，」你說。

「 另外，答案就在你的衣服背後。」