The Rubik’s Cube

Invented in In the mid-1970s by Ernő Rubik (Hungarian) The original (3×3×3) 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.

7

• 8! × 3 ×

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=ylAXYqg bp4M • Closure • Associativity • Inverse • Identity

• INVERSE – Every member of the group must have an inverse. For every g Є G, there is an element g-1 Є G such that

• gº

g-1

=

g-1º g

• ASSOCIATIVE– The order in which the operation is carried out doesn’t matter. • For every g1,g2,g3 Є G, we have • g1º (g2º g3)=(g1º g2)º g3

=e

• IDENTITY – There must exist an element e in the group such that for every g Є G, we have • eºg=gºe=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 Є G, then g1º g2 Є G

Why this applies to the Rubik’s 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 â&#x20AC;˘ In mathematics, computer science, and related subjects, an algorithm is an effective method for solving a problem expressed as a finite sequence of steps. â&#x20AC;˘ Algorithms are used for calculation, data processing, and many other fields.

â&#x20AC;˘ 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 Beginner’s Method…

Only Seven Algorithms… • • • • • • •

1.) 2.) 3.) 4.) 5.) 6.) 7.)

Ri U Fi Ui Ri Di R D U R Ui Ri Ui Fi U F Ui Li U L U F Ui Fi F R U Ri Ui Fi R U Ri U R 2U Ri U R Ui Li U Ri Ui L

Only Six Steps… • • • • • •

1.) 2.) 3.) 4.) 5.) 6.)

Solve the Upper Cross Solve the Upper Corners Solve the Middle Layer Edges Solve the Top Edges Solve the Top Corners Orientate the Top Corners

Let’s 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 you’re 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 it’s spot in the top layer. • Hold the cube with the piece at the lowerfront-right and its home at the upper-frontright, and then do the sequence…

Solving the Upper Corners

Solving the Middle Layer Edges â&#x20AC;˘ 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) â&#x20AC;˘ Step 4 - Solve the Upper Blue Cross â&#x20AC;˘ 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) â&#x20AC;˘ Step 4 - Solve the Upper Blue Cross â&#x20AC;˘ 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 â&#x20AC;˘ 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) â&#x20AC;˘ 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 upperfront-right corner piece. When all the corners have been flipped, just turn the top layer to solve the cube. â&#x20AC;˘ Congratulations, You've done it!

Orientating the Top Corners (Bottom)

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