What do a Rubik’s Cube and a Piano Have in Common?

January 22, 2016 | Elizabeth Knowles

Rubik's cube and a piano.

A nonsensical question with a mathematical answer

In Michael Staff’s Ted-Ed video, he starts off by asking a simple question: How can you play a Rubik’s cube like a piano? Now that question might not really make any sense, but both the puzzle and the musical instrument can be understood through an area of math called group theory.

Group theory is an abstract mathematical field that studies collections of elements that respect four rules — known in mathematics as axioms. A group can be anything from a set of numbers to the orientations of faces on a Rubik’s cube.

Rubik's cube math
Screenshot from Ted-Ed video

Axiom 1: Closure

Any operation performed on an element of a group must lead to another element in the group. If you take the group of integers (negative and positive whole numbers) under addition and, say, add 2+2, you will end up with another integer — 4. The same can be said for the faces of a Rubik’s cube. No matter how you turn the face, you will always end up with an orientation that is part of the group.

Axiom 2 — Associativity:

Regardless of the order in which operations on the elements of the group are performed, you will still end up with the same result. For example, 1+(2+3)=(1+2)+3. In terms of the face of a Rubik’s cube, whether you turn it clockwise one followed by twice, you will end up with the same orientation as if you had turned it twice and then once.

Axiom 3: Identity 

The group must have an element that when added to another element does not change it. For integer addition, that element is 0. For the face of the Rubik’s cube, the identity is not turning the face at all.

Axiom 4: Inverse

Elements in the group are paired such that every element has another element, and when added together, they create the identity element. For example, 4+(-4)=0. Turning the face to the right once and then to the left once brings it back to its original orientation.

SEE ALSO: How to Win at Rock-Paper-Scissors, According to Math

If we think about the Rubik’s cube as a whole rather than looking at a single face, it still satisfies all of the axioms even if we think about turning parts of it in different ways. Each configuration is called a permutation.

Trying to solve a Rubik’s cube randomly is almost impossible, so people who are adept at solving them quickly memorize a list of moves and permutations.

By now, you must be wondering what all of this has to do with a piano. As Staff explains, group theory applies to music as well. A chord can be seen as a group containing the notes that make it up. By permuting the elements, musicians come up with variations on the chord that are also in the group.

The operation that you can perform on a chord is called inversion and involves moving the bottom note to the top. So, if you started with C D# F# A, you now have D# F# A C. Staff says that “composers use inversions to manipulate a sequence of chords and avoid a blocky awkward sounding progression.”

The squares on a Rubik's cube are like musical chords
Screenshot from Ted-Ed video

If you covered the squares on your Rubik’s cube and made sure that every face was a harmonious chord, as you solved it, the corresponding notes would create chords that got more and more harmonious.



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