Group theory 101: How to play a Rubik’s Cube like a piano - Michael Staff

譯者: Adrienne Lin 審譯者: 瑞文Eleven 林Lim

How can you play a Rubik's Cube?

你會如何玩魔術方塊呢？

Not play with it, but play it like a piano?

你可知道玩魔術方塊 與彈鋼琴有相同的原理

That question doesn't make a lot of sense at first,

聽起來似乎不太合理

but an abstract mathematical field called group theory holds the answer,

但是可以在一個名為 群論的數學領域找到答案

if you'll bear with me.

請容我好好解釋

In math, a group is a particular collection of elements.

在數學裡，「群」指的是 特定元素的集合

That might be a set of integers,

可以是一組整數

the face of a Rubik's Cube,

也可以是一組魔術方塊的面板

or anything,

或任何東西

so long as they follow four specific rules, or axioms.

只要他們符合以下四個公理

Axiom one:

公理1：封閉性

all group operations must be closed or restricted to only group elements.

任兩個元素作用後 仍屬於這個集合內

So in our square, for any operation you do,

以魔術方塊為例 無論你怎麼轉動

like turn it one way or the other,

像是讓它朝一邊轉 或另一邊轉

you'll still wind up with an element of the group.

最終結果仍為群內的元素

Axiom two:

公理2：結合律

no matter where we put parentheses when we're doing a single group operation,

當進行操作時 不管將括號放在哪裡

we still get the same result.

結果會依舊相同

In other words, if we turn our square right two times, then right once,

換種說法， 若我們將魔術方塊 順時針轉兩次，然後再轉一次

that's the same as once, then twice,

結果會與順時針轉一次 然後再轉兩次相同

or for numbers, one plus two is the same as two plus one.

以數字來說 1+2=2+1

Axiom three:

公理3：單位元素

for every operation, there's an element of our group called the identity.

「群」必須有一個單位元素

When we apply it to any other element in our group,

當「群」中的任一個元素 與此單位元素作用時

we still get that element.

依舊會得到原本的元素

So for both turning the square and adding integers,

所以對轉方塊或數字加法來說

our identity here is zero,

這個單位元素稱為 0

not very exciting.

聽起來不怎麼特別

Axiom four:

公理4：反元素

every group element has an element called its inverse also in the group.

群中的每一個元素 都必須有一個反元素

When the two are brought together using the group's addition operation,

當這兩個元素一起作用時

they result in the identity element, zero,

都會得到單位元素 0

so they can be thought of as cancelling each other out.

所以他們是被彼此消除的

So that's all well and good, but what's the point of any of it?

一切都很順利 但是這些有什麼關聯呢？

Well, when we get beyond these basic rules,

嗯，在深入這些基本規則後

some interesting properties emerge.

一些有趣的性質漸漸浮現

For example, let's expand our square back into a full-fledged Rubik's Cube.

例如當我們把群論 還原回魔術方塊

This is still a group that satisfies all of our axioms,

將發現它仍然符合四個基本公理

though now with considerably more elements

只是增加了適當的元素

and more operations.

和適當的操作

We can turn each row and column of each face.

我們可以旋轉各面的行及列

Each position is called a permutation,

每種狀態都稱為是一種「排列」

and the more elements a group has, the more possible permutations there are.

一個群有更越多元素 就有更多種排列方式

A Rubik's Cube has more than 43 quintillion permutations,

一個普通魔術方塊裡 排列方式就有4千3百億兆種

so trying to solve it randomly isn't going to work so well.

所以要隨意破解它並不容易

However, using group theory we can analyze the cube

然而，利用群論 我們可以分析魔術方塊的運作

and determine a sequence of permutations that will result in a solution.

而且確定有特別的公式可以破解

And, in fact, that's exactly what most solvers do,

事實上，那就是大部分人所用的技巧

even using a group theory notation indicating turns.

甚至用群論一一解釋

And it's not just good for puzzle solving.

而群論不只有利於解謎

Group theory is deeply embedded in music, as well.

群論也深深嵌入音樂中

One way to visualize a chord is to write out all twelve musical notes

辨識和弦的一個辦法 就是寫出12個音名

and draw a square within them.

然後把他們圍成一圈

We can start on any note, but let's use C since it's at the top.

從任何一個音開始都行 但我先用第一個C來示範

The resulting chord is called a diminished seventh chord.

連成的和弦叫做減七和弦

Now this chord is a group whose elements are these four notes.

那現在這個和弦變成一群 以四個音組成的元素

The operation we can perform on it is to shift the bottom note to the top.

我們可以將最底下的音符移到最上方

In music that's called an inversion,

在音樂上被稱作轉位

and it's the equivalent of addition from earlier.

這與我們之前做的加法是ㄧ樣的

Each inversion changes the sound of the chord,

每次轉位都改變了和弦的聲音

but it never stops being a C diminished seventh.

但它從來不改變減七和弦的本質

In other words, it satisfies axiom one.

換句話說，它滿足公理1

Composers use inversions to manipulate a sequence of chords

作曲家利用轉位來操縱和弦的序列

and avoid a blocky, awkward sounding progression.

並避免不均勻或不和諧的和弦

On a musical staff, an inversion looks like this.

在樂譜上，轉位看起來像這樣

But we can also overlay it onto our square and get this.

我們還可以套用在魔術方塊上 變成這樣

So, if you were to cover your entire Rubik's Cube with notes

如果你將整個魔術方塊 都賦予音名

such that every face of the solved cube is a harmonious chord,

使得魔術方塊的每一面 都是和諧的和弦

you could express the solution as a chord progression

你就可以將解出的方式 以「和弦」的方式呈現

that gradually moves from discordance to harmony

逐漸由不協調轉為和諧

and play the Rubik's Cube, if that's your thing.

如果你喜歡 就來「演奏」魔術方塊吧！