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How can you play a Rubik's Cube?
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Not play with it,
but play it like a piano?
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That question doesn't
make a lot of sense at first,
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but an abstract mathematical field
called group theory holds the answer,
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if you'll bear with me.
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In math a group is a particular
collection of elements.
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That might be a set of integers,
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the face of a Rubik's Cube,
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or anything,
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so long as they follow
four specific rules, or axioms.
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Axiom one:
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all of group operations must be closed
or restricted to only group elements.
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So in our square,
for any operation you do,
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like turn it one way or the other,
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you'll still end up with
an element of the group.
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Axiom two:
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no matter where we put parentheses
when we're doing a single group operation,
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we still get the same result.
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In other words, if we turn our square
right two times, then right once,
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that's the same as once, then twice,
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or for numbers, one plus two
is the same as two plus one.
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Axiom three:
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for every operation, there's an element
of our group called the identity.
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When we apply it
to any other element in our group,
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we still get that element.
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So for both turning the square
and adding integers,
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our identity here is zero,
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not very exciting.
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Axiom four:
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every group element has an element
called its inverse also in the group.
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When the two are brought together
using the group's addition operation,
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they result in the identity element, zero,
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so they can be thought of
as cancelling each other out.
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So that's all well and good,
but what's the point of any of it?
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Well, when we get beyond
these basic rules,
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some interesting properties emerge.
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For example, let's expand our square
back into a full-fledged Rubik's Cube.
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This is still a group
that satisfies all of our axioms,
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though now
with considerably more elements,
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and more operations.
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We can turn each row
and column of each face.
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Each position is called a permutation,
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and the more elements a group has,
the more possible permutations there are.
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A Rubik's Cube has more
than 43 quintillion permutations,
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so trying to solve it randomly
isn't going to work so well.
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However, using group theory
we can analyze the cube
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and determine a sequence of permutations
that will result in a solution.
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And in fact, that's exactly
what most solvers do,
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even using a group theory notation
indicating turns.
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And it's not just good for puzzle solving.
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Group theory is deeply embedded
in music, as well.
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One way to visualize a chord
is to write out all twelve musical notes
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and draw a square within them.
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We can start on any note,
but let's use C since it's at the top.
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The resulting chord is called
a diminished seventh chord.
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Now this chord is a group
whose elements are these four notes.
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The operation we can perform on it
is to shift the bottom note to the top.
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In music that's called an inversion,
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and it's the equivalent
of addition from earlier.
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Each inversion changes
the sound of the chord,
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but it never stops being
a C diminished seventh.
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In other words, it satisfies axiom one.
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Composers use inversions to manipulate
a sequence of chords
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and avoid a blocky,
awkward sounding progression.
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On a musical staff,
an inversion looks like this.
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But we can also overlay it onto our square
and get this.
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So, if you were to cover your entire
Rubik's Cube with notes
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such that every face of the solved cube
is a harmonious chord,
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you could express the solution
as a chord progression
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that gradually moves
from discordance to harmony
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and play the Rubik's Cube,
if that's your thing.