Can you solve the cursed dice riddle? - Dan Finkel

譯者: Lilian Chiu 審譯者: 穎君 紀

Ah, spring.

啊，春天。

As Demeter, goddess of the harvest, it’s your favorite season.

你是豐收女神狄蜜特， 而這是你最愛的季節。

Humans and animals look to you to balance the bounty of the natural world,

人類及動物都期待 你會平衡自然界的豐饒，

which, like any self-respecting goddess, you do with a pair of magical dice.

而你和所有自重的女神一樣， 用一對魔法骰子來做到這件事。

Every day you roll the dice at dawn,

每天，你會在黎明時分擲骰子，

and all lands that match the sum of the two dice produce their resources.

凡是對應兩個骰子數字總和的 土地都會生產出其資源。

The resulting frequency of sums across the season

整季中各總和數字出現的次數

keeps your land in perfect harmony; any other rates would spell ruin.

讓你的土地能維持完美的和諧；

其他的比率都會毀掉土地。

And that’s why it was particularly rotten when Loki, the Norse trickster god,

所以有件事會讓你非常苦惱：

北歐的惡作劇之神洛基

invaded your land and cursed your dice, causing all the dots to fall off.

入侵你的土地，詛咒你的骰子，

讓骰子上所有的圓點都掉落下來。

When you try to reaffix them,

當你試著把圓點黏回去時，

you find that one die won’t accept more than four dots on any of its sides,

你發現有一個骰子

不接受任何一面有 超過四個圓點在上面，

though the other has no such constraint.

另一個骰子就沒有這項限制。

You can use Hephaestus’ furnace to seal the dots in place before the sun rises,

你可以在太陽升起前

用赫菲斯托斯的工匠爐 把圓點固定上去，

but once sealed you can’t move or remove them again.

但固定之後，就無法 再移動或移除任何圓點了。

How can you craft your dice so that, when rolled and summed,

你要怎麼製作你的骰子，才能 讓擲骰子之後把數字加總，

every total comes up with the exact same frequency

各總和出現的次數剛好符合

as it would with standard 6-sided dice?

標準的六面骰子擲出的結果？

Pause here to figure it out for yourself. Answer in 3

若想自行解題，請在此暫停。 答案即將公佈：三。

Answer in 2

答案即將公佈：二。

Answer in 1

答案即將公佈：一。

Normal dice can roll any sum from 2 to 12,

正常的骰子可以擲出二 到十二之間的任何總和。

but sums in the middle tend to come up more frequently than ones on the ends.

但中間的總和會比兩端的更常出現。

We can see the odds of rolling any sum by making a table,

我們可以製作一個表格來呈現 擲出任何總和的機率，

with all the possibilities for one die represented on the top,

一個骰子的所有可能性列在上方，

and those for the other on the side.

另一個骰子的則在側邊。

The table lets us see at a glance that there are six ways to roll a 7,

這個表格讓我們一看就知道 有六種方式可以擲出七，

but only two ways to roll a 3.

但只有兩種方式可以擲出三。

This also gives us an approach to crafting our new set of dice.

這個表格也讓我們知道怎麼作新骰子。

Matching the original sum frequencies means that the locations of the sums

要符合原本的次數，

意味著這個表格中 總和的位置可能會改變，

in this table may change,

but the numbers and quantities of each sum must stay the same.

但每個總和數值和數量 都要和原本一樣。

In other words, there still must be exactly one 2, two 3s, and so on.

換言之，仍然必須要剛好有一個二、

兩個三，以此類推。

To start, we’ve got to roll that 2,

一開始，我們得要擲出二，

and since we have to use positive, whole numbers,

因為我們必須要用正整數，

there’s only one choice: each die needs a 1 on it.

只有一個選擇，每個骰子 都得有一面是一。

What else do we know?

我們還知道什麼？

Assuming we have a 4— the highest number possible— on the cursed die,

假設我們有一面是四——

被詛咒的骰子上最高就只能有四，

the other one would need an 8 in order to have one way to roll 12.

另一個骰子就得有一面是八，

才能有一種方式擲出十二。

In fact, we know we require a single 1 and a single 4 on the cursed die,

事實上，我們知道被詛咒的骰子上 只能有一面是一、一面是四，

or we’d have too many ways to roll a 2 or a 12.

若不是這樣，就會有太多 方式可以擲出二或十二。

So the cursed dies remaining four sides must have a mix of 2s and 3s.

所以，被詛咒的骰子所剩下的四面

一定是二或是三。

If we have three or four 2s, we can roll the sum 3 too many ways.

如果三面或四面都是二，

就有太多方式可以擲出三。

Similarly, if we have three or four 3s, we’d get the sum 11 too often.

同樣的，如果有三面或四面是三，

就會太常擲出十一。

So the only possibility is for the cursed die to have two 2s and two 3s.

所以唯一的可能性 就是讓被詛咒的骰子

有兩面是二、兩面是三。

With one die completed,

一個骰子完成了，

we should be able to figure out the missing values on the second.

我們應該就能想通第二個骰子 還空白的那幾面會是多少。

First, we need one more way to make 10 and 4,

首先，我們還需要一種方式 擲出十，一種擲出四。

so we must have one 3 and one 6.

所以一定有一面是三、一面是六。

We now need one more way to make 5 and 9.

現在還需要一種方式 擲出五、一種擲出九。

That forces us to choose 4 and 5 for the final sides.

我們被迫選擇四和五做為最後兩面。

Fill them in, and lo and behold,

把骰子做好，接著，看哪，

we have a distribution table where every possible sum

我們得到的次數分配表格中

每個可能的總和出現的次數

shows up the same number of times as with our original dice!

都和原始的骰子一樣！

The discovery of these dice was made in 1978 by George Sicherman,

薛克曼在 1978 年發現了這組骰子，

which is why they’re referred to as “Sicherman dice.”

這就是為什麼它們被稱為「薛骰」。

Incredibly, this is the only alternate way to make two 6-sided dice

針對兩個六面骰子，

只有這一種替代方式

with the same distribution of sums as standard dice.

能產生出和標準骰子完全 一樣的總和次數分佈。

You send the dice to be reforged, confident that you’ve averted disaster.

你把骰子送去重新鑄造，

確信自己成功阻止災難發生。

Now it’s time to repay the Norse gods with a gift of your own.

現在該換你準備禮物 來報答這些北歐的神了。