The Rubiks Cube

Invented in In the mid-1970s by Ern Rubik (Hungarian)

The original (333) Rubik's Cube has eight corners and

twelve edges. (made up of little cubies)

Combinations (Permutations)

There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the

orientation of the eighth depends on the preceding seven, giving 37(2,187) possibilities.

There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well.

Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.

8! 37 12!

2 211 =

There are exactly 43,252,003,274,489,856,000 permutations, which is approximately forty-three quintillion.

(approximately 4.33 x 1019)

(BY TWISTING ALONE - NOT BY DISASSEMBLY)

Group Theory

It is the study of symmetry

http://www.youtube.com/watch?v=ylAXYqgbp4M

Closure

Associativity

Inverse

Identity

INVERSE Every member of

the group must have an

inverse. For every g G,

there is an element g-1 G

such that

g g-1 = g-1 g = e

ASSOCIATIVE The order in

which the operation is carried

out doesnt matter.

For every g1,g2,g3 G, we have

g1 (g2 g3)=(g1 g2) g3

IDENTITY There must

exist an element e in the

group such that for every

g G, we have

e g = g e = g

CLOSURE If two elements

are members of the group

(G), then any combination of

them must also be a member

of the group. For every g1,g2 G, then g1 g2 G

Why this applies to the Rubiks Cube

Closure yes, whatever moves are carried out it still remains a cube.

Associativity yes (FR)L=F(RL).

Identity yes, by doing nothing.

Inverse yes, by doing the moves backwards you get back to the identity, eg

(FRBL)(L-1B-1R-1F-1)=e

Therefore we have a group.

Algorithms

In mathematics, computer science, and related subjects, an algorithm is an effective method for solving a problem expressed as a finite sequence of steps.

Algorithms are used for calculation, data processing, and many other fields.

Algorithms are a list of well-defined instructions for completing a task. Starting from an initial state, the instructions describe a computation that proceeds through a well-defined series of successive states, eventually terminating in a final ending state.

Cube Notation

*Each side of the cube is represented by a letter

The Beginners Method

Only Seven Algorithms

1.) Ri U Fi Ui

2.) Ri Di R D

3.) U R Ui Ri Ui Fi U F

4.) Ui Li U L U F Ui Fi

5.) F R U Ri Ui Fi

6.) R U Ri U R 2U Ri

7.) U R Ui Li U Ri Ui L

Only Six Steps

1.) Solve the Upper Cross

2.) Solve the Upper Corners

3.) Solve the Middle Layer Edges

4.) Solve the Top Edges

5.) Solve the Top Corners

6.) Orientate the Top Corners

Lets Begin

Solving the Upper Cross

Step 1 - Solve the Upper Green Cross

HINT: To solve the green cross, you have to solve each green edge piece on your own, one by one

The tricky part is not messing up the ones youre already solved. You have to figure this part out for yourself.

Solving the Upper Cross

Solving the Upper Corners

HINT: Find a corner piece in the bottom layer that belongs on top. Turn the bottom layer until that piece is directly below its spot in the top layer.

Hold the cube with the piece at the lower-front-right and its home at the upper-front-right, and then do the sequence

Solving the Upper Corners

Solving the Middle Layer Edges

HINT: Now flip the cube over so green is on the bottom. Try to find the red-yellow edge piece. If its in the top layer, turn it until the edge matches one of the pictures on the right. Then do the corresponding sequence to solve it. If the red-yellow edge piece is somewhere in the middle layer, but its in the wrong place or flipped the wrong way, hold the cube so that the red-yellow edge is in the front-right position, and do either sequence once. (This may require you to rotate the cube to a new face). After the move, the piece is in the top layer, and you can solve it as described above. Repeat this for the other three middle-layer edges.

Solving the Middle Layer Edges

Solving the Top Edges (Bottom)

Step 4 - Solve the Upper Blue Cross

HINT: Turn the top layer until the edges match one of these pictures. If you do the sequence on the right once and you still don't have a blue cross, then repeat this step until you do. It doesn't matter which face you start with. Note: In this step, there will be other blue pieces showing on your cube that do not appear in these diagrams.

Solving the Top Edges (Bottom)

Solving the Top Edges (Bottom)

Step 4 - Solve the Upper Blue Cross

HINT: Hold the cube with red in front. Turn the top layer until the red and blue edge piece is solved as in the picture, and then repeat the sequence on the right until the yellow and blue edge piece is also solved, on the right side. Now turn the whole cube so that white is the 'Front' face. If the top white edge isn't solved, just do the sequence once more, followed by 'U' to position all the edges properly.

Solving the Top Edges (Bottom)

Solving the Top Corners

HINT: Find a corner piece that's in the right place, and hold the cube with that piece above your right thumb. In the picture, this piece is the blue, yellow, and red piece. Don't turn the top layer at all, because you will mess up the edges that you just solved in step 5. Now do the sequence on the right once or twice to put the other 3 corners in to the right places. If you can't find a corner piece in the right place, just do the sequence on the right once before you start this step.

Solving the Top Corners

Orientating the Top Corners (Bottom)

HINT: Hold the cube with red in front. Keep turning the top layer until the upper-front-right corner needs to be flipped, to have blue on top, like in the picture. Do the sequence on the right either 2 or 4 times to flip the corner so that blue is on top. Note: As you work through this step, lower layer colours may become scrambled. Don't worry, just keep going! With red still in front, keep turning the top layer and do the sequence again whenever needed to flip the upper-front-right corner piece. When all the corners have been flipped, just turn the top layer to solve the cube.

Congratulations, You've done it!

Orientating the Top Corners (Bottom)