The Rubik’s Cube

The History of Rubik Cube

Created in 1974 by a Professor of architecture named Erno Rubik

This was suppose to be an object that was not possible. It consists of 26 cubes

The solid cube twisted and turned without breaking

On each side of the cube are colorful stickers which gets scrambled hence the “Rubik’s Cube”

It took Erno over a month to find a solution to this puzzle

Popularity

The Rubik’s Cube was released in the world market in 1980

There was great international interest in the Rubik’s Cube

Competitions were held for fastest “cubist”

The first world Champion took place on June 5, 1982 in Budapest

Today much of the craze has faded

Know Your Rubik’s Cube

Edge pieces have two colors (12) Corner pieces have three colors (8) Center pieces have 1 color- They do

not move and represent the color of their side

White is opposite Yellow Orange is opposite Red Green is opposite Blue

Permutations

The number of possible permutations for a Rubik’s Cube are:

8 corner pieces can be arranged 8! Ways, each of which can be arranged in 3 orientations thus 3^8 possibilities for each permutation

12 edge pieces can be arranged 12! Ways, each side piece has 2 orientations thus 2^12 arrangements

Cont..

Only 1/3 of the permutations have the rotation od corner pieces correct

Only ½ of permutations have the correct orientation for the side pieces

Only ½ have the correct orientation in general

possible arrangements, but only one correct solution!

Rotations of 90,180,270 degrees of the front, right, left, right, upper, lower and back faces are used

From these rotations, algorithms have been created to solve the Rubik’s Cube

Group Theory Rubik's cube can be viewed as a group, where each

element of the group is a permutation. As a group, it has the following properties:

Closure:If P1 and P2 are two permutations in the group, then P1P2 is also a permutation in the same group

Associativity Performing P1 followed by P2P3 is the same as performing P1P2 followed by P3. Identity:There is a permutation in the group in which no pieces are moved. Inverse:For each permutation in the group, there exists an inverse permutation which has the reverse effect.

Rubik's Cube also has a number of subgroups, each having these same 4 properties.

Solving the Rubik’s Cube

Step 1: getting a white cross with a yellow center

Step 2: get your white cross centered at the white piece.

This is done by looking at one of your white side pieces and looking at the color on it’s side, and making a 180 degree rotation.

Step 3: get your white corner pieces to have a completed white side

Step4: Solve for the second row. This is done by turning the solve white side to the back and having the yellow side face you. You look at all the side pieces that DON’T have yellow and you use the permutation (F-L-R-C-Rcc) (if your going right to left)and then fix your white side. Same Idea if you are going left to right

https://www.youtube.com/watch?v=rmnSpUgOvyI (7:26)

Step 5: This step depends on the pattern that you acquire on the bottom of your Rubik’s Cube.

Step 1 Step 2 Step3 Step 4

State 1 is fine but for state 2-4 there are different algorithms you need to do

State 2 : F U R U' R' F‘ State 3 :  F R U R' U' F‘ State 4 : Either algorithm for

state 2 or 3 should work After these steps you should be

left with a yellow cross (state 1)

Step 6: Here you solve for the yellow corners

For this you will do the algorithm R,U,R’,U,R,2U,R’ twice.

Once you have the face looking like this you do the above algorithm again

Once done you will have the white side solved, the yellow side solved, and the first two layers.

Final two steps solving for the corners and the last side piece

https://www.youtube.com/watch?v=rmnSpUgOvyI (18:22)

Exam Question

How many arrangements of the Rubik’s Cube can you have?

Answer:

References

http://www.rubiks.com/history http://www.math.ubc.ca/~cass/courses/m3

08/projects/rtran/rtran.pdf

https://www.youtube.com/watch?v=rmnSpUgOvyI

http://ruwix.com/the-rubiks-cube/how-to-solve-the-rubiks-cube-beginners-method/

http://www.ryanheise.com/cube/theory.html