Solving the Rubik's Cube Puzzle

Step-by-step solution in a couple of YouTube videos by Dan Brown.

I just signed-up to YouTube yesterday, and this post is mostly meant as practice for myself posting videos into my blog.
While exploring YouTube’s archives I found a few videos about solving the Rubik’s Cube puzzle. Rubik’s cube is one of my favorites puzzles because it is closely related to both Group Theory and Graph Theory, branches of modern math. Playing with Rubik’s cube also helps somehow develop one’s intuition about the Cartesian (x, y, z) coordinate system in 3-D space.
In the two videos below, Dan Brown incorporates a little algebraic notation to precisely describe a few sequences he uses in his general solution of the Rubik’s cube.
So far I have not used Rubik’s cube as a teaching aid in any of my tutoring sessions, so this post really does not necessarily have a lot to do with tutoring but I decided to include it anyway because the puzzle does have to do with math, and it is fun.
I hope you will enjoy the videos!

P.S. After loading these first videos I decided to search for other videos with content related to that of my previous posts, so I will be including some more videos in those older posts too.

Solving Equations For a Particular Variable

A very basic principle

An equation has one equal sign.

The equal sign divides the equation into left hand side and right hand side.

The two sides may look totally different from each other as expressions but the equal sign says their numerical value has to be the same.

The fundamental principle of equations says that, when two expressions have the same numerical value, if we apply one operation to both expressions, the resulting expressions after the operation is performed will also be equal in value. They will be equal not to the original expressions, but to each other.

So, if A, B and C are three algebraic expressions, and we have the equation A = B, then all of the following will also be valid equations:

A + C = B + C

A - C = B - C

(A)(C) = (B)(C)

A/C = B/C [provided C is not zero]

A^2 = B^2

Square root of A = Square root of B

This fundamental principle is used over and over to solve equations for specific variables, one step at a time.

For example, in solving for x the equation (3x + 1)/2 = 5y - 4, we can do it like this:

1) Multiply both sides by 2 and we get

3x + 1 = 2(5y - 4)

2) Subtract 1 from both sides and we get

3x = 2(5y - 4) - 1

3) Divide both sides by 3 and we get

x = ( 2(5y - 4) - 1)/3

Now the equation has been solved for x in a series of steps, where each step consists of applying one and the same operation to BOTH sides of the equation.

The fact that the resulting expression for x can be simplified to

x = (10y -9)/3

is not relevant here. I am only illustrating the process we use to isolate x one step at a time by applying the same operation to both sides of the equation.

The following YouTube video from InterAlgebra12 shows several more examples: