Britney: Hello and welcome to (The Hidden Algebra of the Rubiks Cube).
I’m your host Britney and I’m here with my co-hosts
Kyra + Youssef
Britney: Here we will talk about group theory and its application to the Rubik’s cube.
Youssef: In today’s episode we will discuss the history of how this famous toy came to be.
Kyra: In 1974, a Hungarian architect by the name of Erno Rubik created the first Rubik's cube. He originally called it “Buvos Kocka,” which translates to magic cube. Rubik created this toy while teaching at the Academy of Applied Arts with the intention of showing his students how to model three-dimensional movements. To do this, he spent months tinkering with wooden blocks. Then boom!!
Britney: He created the first Rubik's cube. An agent then pitched this idea to create a toy to a company and soon the puzzle became extremely popular selling more than 350 million cubes as of 2018
Youssef: The cube had a large influence on pop culture and created a competitive sport called speedcubing. While many people have a Rubik's cube, most can’t solve it.
Kyra: That might not surprise you, considering there are more than 43 quintillion possible permutations! To master the cube you must learn a sequence of movements and apply them in a particular order. If we want to understand these sequences we need to use an application of math called
Briney Group theory, which we will cover in our next episode of
All ( The Hidden Algebra of the Rubiks Cube)
Episode 2: Group Theory
Britney: Hello and welcome back to another episode of (The Hidden Algebra of the Rubiks Cube)
Kyra: As you recall in the last episode we covered a brief history on the rubik's cube.
Youssef: In today's episode we will pick up where we left off and explain what group theory is.
Kyra: To understand the sequences, rotations and variations of a rubik's cube we first need to explain what group theory is
Britney: Group theory is a collection of elements or objects put together for an operation.
Youssef: To be a group, certain properties need to be present. A group must have
associativity, inverses, an identity, and closure.
Kyra: Group theory loves the idea of symmetry between sets. The Rubik’s cube carries 27 ‘cubies’ of which 26 are actually visible.
Britney: The small cubes or ‘cubies’ are rotated on their axis in arrays of 3 where each array is rotated 180, 90, and 360 degrees.
Youssef: When each array is rotated, they’re still symmetric, tying symmetry and group theory together.
Kyra: In the “Becoming a Whole Mathematician” seminar, our guest speaker Dandrielle Lewis (an inspiring mathematician) showed a way in which symmetry and group theory come together with a butterfly.
Britney: Let’s say you’ve drawn a perfect butterfly where both sides are congruent. When you fold the butterfly. It remains the same. Even when rotated, it still remains the same. Therefore, group theory can be applied to various things.
Youssef: A concept group theory is applicable to is algebraic structures and permutations which we will further discuss in our next episode of
All (The Hidden Algebra of the Rubik’s Cube)
Episode 3: Symmetry and Permutations
Britney: Hello and welcome back to another episode of (The Hidden Algebra of the Rubik’s Cube)
Kyra: In today's episode we will discuss symmetry and permutations.
Youssef: A permutation is one way that a set of numbers or objects can be arranged. Within the Rubik’s cube, there is a group that is composed of permutations known as the symmetric group.
Kyra: In terms of a cube there are 24 rotational symmetries.
Britney: Within this symmetric group, a permutation switches between 2 numbers. So, if we have (2,3) and (1,2), this would leave us with (1,2,3).
Kyra: Rings, fields, and vector spaces can be seen within these groups with additional operations and axioms. Algebraic structures also include these groups.
Youssef: An algebraic structure is a set of numbers, objects, or otherwise with at least one finite operation defined on it that satisfies a list of axioms. Axioms are rules accepted to be true no matter the type of actions that its placed upon them. These axioms have a set of rules that each operation follows that we can see in action on the Rubik’s cube.
Kyra: We will discuss this more in our next episode of (The Hidden Algebra of the Rubik’s Cube)
Episode 4: Step by Step Mathematical Operations
Britney: Hello and welcome to the final episode of (The Hidden Algebra of the Rubik’s Cube) 6
Kyra: Referencing one of our first episodes, a group is a collection of elements with an operation that has associativity, inverses, an identity, and closure.
Kyra: As you know the goal of this puzzle is to move all of the edge cubies and corner cubies into their correct positions with their correct orientations
Youssef: This means that there needs to be some symmetry so that everything is oriented correctly.
Britney:There are many possible ways to permute a Rubik’s cube. This number is represented by
Youssef: 12 factorial ways to permute the edge cubies
Kyra: 8 factorial ways to permute the corners
Britney: 2^12 ways to orient (or align) the edge “cubies”
Youssef: 3^8 ways to orient the corner “cubies”
Youssef: So our equation would be 12!8!2^123^8…. Which equals 519 quintillion, 024 quadrillion, 039 trillion, 293 billion, 878 million, 272 thousand, 000. This is more permutations than the number of grains of sand on all of earth’s beaches!
Youssef: It may also be important to understand the notation used when learning to solve the Rubik's cube. Each letter corresponds to a clockwise quarter-turn. F means front, B means back, U means upper, D means lower, R means right, and L means left.
Kyra: The most common method to solve the Rubik’s cube is by first solving the lower two layers of the cube, then aligning the “cubies” on the top layer so that the top face of the cube is the correct color. Finally, you permute the “cubies” on the top layer so that they are in their correct positions.
Britney: To combine moves on a Rubik’s cube, there are some steps to follow:
Lets say X and Y are two moves on the Rubik’s cube
So, when you do XY this means “do, X then do Y”
X2 would mean “repeat X twice”
Youssef: If two moves are made on the cube that have the same effect, they are considered equal.
Britney: Inverses are another idea used with the Rubik’s cube which will be notated as X’ which means to turn that face counterclockwise
Britney: A commutator in a Rubik’s cube measures the failure of X and Y being commutative since a Rubik’s cube you cannot always use the commutative property
Youssef: An example that proves a failure to this identity is when you flip two edges and rotate two corners
Youssef: Conjugation is another way we can use to build up new moves to solve Rubik’s cubes
Kyra: For example, let’s say that we have X and Y and we decide to bring in another new move noted as Z.. this would be known as the conjugate of X.
This identity does nothing to your original orientation.
Youssef: Any two symmetries of the same type are conjugates!
Britney: On the cube we have some examples of groups that don’t commute.
Such as six 180 degree rotations about a line through the centers of two opposite edges. Eight 120 degree rotations about a line through two opposite corners.
Youssef: Six 90 degree rotations about a line through the centers of two opposite faces. Three 180 degree rotations about a line through the centers of two opposite faces.
Kyra: All of these examples form a nonabelian group which means that at least one pair of elements in a set doesn’t commute.
Britney: That just about wraps up this episode and series of The Hidden Algebra of the Rubik’s Cube.
Youssef: Thank you for listening, and
All: Have fun learning to solve the cube !!