Can you solve the cursed dice riddle? - Dan Finkel

877,800 views ・ 2023-09-07

TED-Ed


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譯者: Lilian Chiu 審譯者: 穎君 紀
00:07
Ah, spring.
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啊,春天。
00:08
As Demeter, goddess of the harvest, it’s your favorite season.
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你是豐收女神狄蜜特, 而這是你最愛的季節。
00:12
Humans and animals look to you to balance the bounty of the natural world,
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人類及動物都期待 你會平衡自然界的豐饒,
00:17
which, like any self-respecting goddess, you do with a pair of magical dice.
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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。
00:22
Every day you roll the dice at dawn,
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每天,你會在黎明時分擲骰子,
00:25
and all lands that match the sum of the two dice produce their resources.
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凡是對應兩個骰子數字總和的 土地都會生產出其資源。
00:30
The resulting frequency of sums across the season
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整季中各總和數字出現的次數
00:33
keeps your land in perfect harmony; any other rates would spell ruin.
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讓你的土地能維持完美的和諧;
其他的比率都會毀掉土地。
00:38
And that’s why it was particularly rotten when Loki, the Norse trickster god,
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所以有件事會讓你非常苦惱:
北歐的惡作劇之神洛基
00:43
invaded your land and cursed your dice, causing all the dots to fall off.
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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。
00:49
When you try to reaffix them,
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當你試著把圓點黏回去時,
00:51
you find that one die won’t accept more than four dots on any of its sides,
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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,
00:57
though the other has no such constraint.
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另一個骰子就沒有這項限制。
01:00
You can use Hephaestus’ furnace to seal the dots in place before the sun rises,
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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,
01:05
but once sealed you can’t move or remove them again.
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但固定之後,就無法 再移動或移除任何圓點了。
01:10
How can you craft your dice so that, when rolled and summed,
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你要怎麼製作你的骰子,才能 讓擲骰子之後把數字加總,
01:14
every total comes up with the exact same frequency
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各總和出現的次數剛好符合
01:17
as it would with standard 6-sided dice?
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標準的六面骰子擲出的結果?
01:21
Pause here to figure it out for yourself. Answer in 3
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若想自行解題,請在此暫停。 答案即將公佈:三。
01:23
Answer in 2
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答案即將公佈:二。
01:26
Answer in 1
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答案即將公佈:一。
01:29
Normal dice can roll any sum from 2 to 12,
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正常的骰子可以擲出二 到十二之間的任何總和。
01:33
but sums in the middle tend to come up more frequently than ones on the ends.
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但中間的總和會比兩端的更常出現。
01:38
We can see the odds of rolling any sum by making a table,
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我們可以製作一個表格來呈現 擲出任何總和的機率,
01:42
with all the possibilities for one die represented on the top,
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一個骰子的所有可能性列在上方,
01:46
and those for the other on the side.
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另一個骰子的則在側邊。
01:48
The table lets us see at a glance that there are six ways to roll a 7,
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這個表格讓我們一看就知道 有六種方式可以擲出七,
01:53
but only two ways to roll a 3.
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但只有兩種方式可以擲出三。
01:55
This also gives us an approach to crafting our new set of dice.
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這個表格也讓我們知道怎麼作新骰子。
02:00
Matching the original sum frequencies means that the locations of the sums
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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,
02:05
in this table may change,
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02:07
but the numbers and quantities of each sum must stay the same.
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但每個總和數值和數量 都要和原本一樣。
02:11
In other words, there still must be exactly one 2, two 3s, and so on.
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換言之,仍然必須要剛好有一個二、
兩個三,以此類推。
02:17
To start, we’ve got to roll that 2,
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一開始,我們得要擲出二,
02:20
and since we have to use positive, whole numbers,
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因為我們必須要用正整數,
02:22
there’s only one choice: each die needs a 1 on it.
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只有一個選擇,每個骰子 都得有一面是一。
02:26
What else do we know?
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我們還知道什麼?
02:28
Assuming we have a 4— the highest number possible— on the cursed die,
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假設我們有一面是四——
被詛咒的骰子上最高就只能有四,
02:32
the other one would need an 8 in order to have one way to roll 12.
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另一個骰子就得有一面是八,
才能有一種方式擲出十二。
02:37
In fact, we know we require a single 1 and a single 4 on the cursed die,
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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,
02:43
or we’d have too many ways to roll a 2 or a 12.
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若不是這樣,就會有太多 方式可以擲出二或十二。
02:47
So the cursed dies remaining four sides must have a mix of 2s and 3s.
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所以,被詛咒的骰子所剩下的四面
一定是二或是三。
02:54
If we have three or four 2s, we can roll the sum 3 too many ways.
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如果三面或四面都是二,
就有太多方式可以擲出三。
02:58
Similarly, if we have three or four 3s, we’d get the sum 11 too often.
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同樣的,如果有三面或四面是三,
就會太常擲出十一。
03:04
So the only possibility is for the cursed die to have two 2s and two 3s.
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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。
03:10
With one die completed,
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一個骰子完成了,
03:12
we should be able to figure out the missing values on the second.
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我們應該就能想通第二個骰子 還空白的那幾面會是多少。
03:16
First, we need one more way to make 10 and 4,
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首先,我們還需要一種方式 擲出十,一種擲出四。
03:19
so we must have one 3 and one 6.
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所以一定有一面是三、一面是六。
03:23
We now need one more way to make 5 and 9.
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現在還需要一種方式 擲出五、一種擲出九。
03:26
That forces us to choose 4 and 5 for the final sides.
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我們被迫選擇四和五做為最後兩面。
03:31
Fill them in, and lo and behold,
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把骰子做好,接著,看哪,
03:33
we have a distribution table where every possible sum
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我們得到的次數分配表格中
每個可能的總和出現的次數
03:36
shows up the same number of times as with our original dice!
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都和原始的骰子一樣!
03:40
The discovery of these dice was made in 1978 by George Sicherman,
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薛克曼在 1978 年發現了這組骰子,
03:46
which is why they’re referred to as “Sicherman dice.”
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這就是為什麼它們被稱為「薛骰」。
03:50
Incredibly, this is the only alternate way to make two 6-sided dice
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針對兩個六面骰子,
只有這一種替代方式
03:55
with the same distribution of sums as standard dice.
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能產生出和標準骰子完全 一樣的總和次數分佈。
03:59
You send the dice to be reforged, confident that you’ve averted disaster.
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你把骰子送去重新鑄造,
確信自己成功阻止災難發生。
04:04
Now it’s time to repay the Norse gods with a gift of your own.
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現在該換你準備禮物 來報答這些北歐的神了。
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