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

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Mathematics explains the workings of the universe, from particle physics to engineering and economics. Math is even closely related to music, and their common ground has something to do with a Rubik's Cube puzzle. Michael Staff explains how group theory can teach us to play a Rubik’s Cube like a piano.

Lesson by Michael Staff, animation by Shixie.

How can you play a Rubik's Cube?

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

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

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

if you'll bear with me.

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

That might be a set of integers,

the face of a Rubik's Cube,

or anything,

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

Axiom one:

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

So in our square,
for any operation you do,

like turn it one way or the other,

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

Axiom two:

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

we still get the same result.

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

that's the same as once, then twice,

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

Axiom three:

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

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

we still get that element.

So for both turning the square
and adding integers,

our identity here is zero,

not very exciting.

Axiom four:

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

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

they result in the identity element, zero,

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

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

Well, when we get beyond
these basic rules,

some interesting properties emerge.

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

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

though now
with considerably more elements

and more operations.

We can turn each row
and column of each face.

Each position is called a permutation,

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

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

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

However, using group theory
we can analyze the cube

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

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

even using a group theory notation
indicating turns.

And it's not just good for puzzle solving.

Group theory is deeply embedded
in music, as well.

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

and draw a square within them.

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

The resulting chord is called
a diminished seventh chord.

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

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

In music that's called an inversion,

and it's the equivalent
of addition from earlier.

Each inversion changes
the sound of the chord,

but it never stops being
a C diminished seventh.

In other words, it satisfies axiom one.

Composers use inversions to manipulate
a sequence of chords

and avoid a blocky,
awkward sounding progression.

On a musical staff,
an inversion looks like this.

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

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

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

you could express the solution
as a chord progression

that gradually moves
from discordance to harmony

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