Solving the Rubik

1
Solving the Rubiks Cube
Based on The beginners solution to the Rubiks
cube by Jasmine Lee
2
Overview

• We can look at the Rubiks cube as having 3
  levels, and imagine 9 little cubes on each level.
  So we can imagine 27 little cubes that make up
  the whole Rubiks cube. Well call the little
  cubes cubies.
• The cubie at the very center of the cube is never
  visible and there is no motion of the cube that
  will ever reveal it. In fact, of course, there
  is no cubie at the center of the cube, there are
  really just 26 cubies that we can see.
• These 26 visible cubies are of three different
  types corners, edges, and centers.

3
Overview

• Notice that with any motion of the cube, the
  center cubie on each face stays put. That is,
  the center cubie on each face is fixed relative
  to the other center cubies on the other faces.
  Well call these 6 cubies center cubies
  (referring to the center of each face). We use
  them to identify which is the red, green, blue,
  white, orange, and yellow side.
• There are 8 corner cubies. These are the cubies
  with 3 visible faces (called facelets) Notice
  how they will always stay corner cubies after any
  motion of the cube.
• There are 12 edge cubies. These are the cubies
  with 2 facelets and are always between the corner
  cubies. Notice how edge cubies remain edge cubies
  after any motion of the cube.
• Now we have 6 centers 8 corners 12 edges that
  make 26 visible cubies.

4
Overview

• The process of solving the cube can be done in 3
  stages the first (top) layer, middle (second)
  layer and then the last (bottom) layer. These
  layers correspond to the top 9 cubies, the middle
  8 (visible) cubies and the bottom 9 cubies.
• In each stage there are certain algorithms that
  complete that stage. To understand these
  algorithms, you will need to know some notation
• When you look at a cube, there is always a top,
  bottom, right, left, front and back. We call
  these sides Up, Down, Right, Left, Front and
  Back. Be careful not change the position of the
  cube or these will change!
• All of the motions of the cube are clockwise or
  counter-clockwise turns of the 6 faces. Be
  careful when you do clockwise and
  counter-clockwise turns of the down and back
  sides. We take clockwise to be from the
  perspective of someone looking directly at that
  face.

5
Overview

• For most of this guide we will use the white side
  of the cube as the up side.
• For clockwise turns of each side we use the
  capital letters U,D,F,L,R,B and for
  counter-clockwise turns we use U, D, F,L,R,
  and B.
• Also, as a shortcut, we will write U2 to mean two
  clockwise turns of the up side.
• For example, the sequence L U' R' U L' U' R U2
  means
• Turn the left side clockwise, the up side
  counter-clockwise, the right-side
  counter-clockwise, the up side clockwise, the
  left-side counter-clockwise, the up-side
  counter-clockwise, the right side clockwise and
  finally the up-side clockwise twice.
• Did you read all that? Now you see the advantage
  of the shorthand notation!

6
Overview

• To do the cube here are the algorithms you will
  need to know...
• For the middle layer
• 1) D L D' L' D' F' D F
• 2) D' F' D F D L D' L'
• For the last layer
• 3) F U R U' R' F
• 4) L U' R' U L' U' R U2
• 5) R U R' U R U2 R' U2
• 6) R2 U F B' R2 F' B U R2
• 7) R2 U' F B' R2 F' B U' R2
• Of course, you will also need to know when to use
  them

7
The First Layer

• The best thing is to learn the first layer on
  your own. If you can
• do this, youll have a little more confidence and
  comfort with
• how the cube works when we do the middle and last
  layer.
• But here are some hints
• To do the first layer, we do two things
• Form the white cross
• Put the corners in place
• When you make the cross, you must line up each
  edge cubie
• in the cross with the matching center of each
  adjacent face.
• The edges are correctly placed in the diagram
  above and
• incorrectly in the diagram below it.
• Making the white cross means placing 4 edge
  cubies in their
• correct locations with respect to the adjacent
  centers.

Correctly placed edges
These edges are placed incorrectly
8
The First Layer
To get the white cross, find all edge cubies with
a white face. Do one edge cubie at a time.
For example, suppose you want to place the
blue/white edge into the cross.
Correctly placed edges
Rotate the blue/white edge so that the blue
facelet matches up correctly with the blue
center cubie. Then rotate the blue side until
the white facelet of the blue/white edge is in
the cross. Now do this for each of the edges
with a white facelet. Often, youll rotate an
edge along the bottom layer to match it with the
correct center, then rotate it up into the cross.
It can happen that placing one edge of the cross
into position will move another one you had
previously placed out of position. Dont worry,
it wont go far. Youll find that it sometimes
happens, but it is always a single quarter (90
degree) turn to put the edge back in
place. Anyway, with this layer, you just need to
take some time to get to know how to permute the
corners and edges.
9
The First Layer
Everything about the middle and last layer
assumes the first layer is done correctly. Each
corner must be placed as shown in the diagram, so
its facelets match the color of the center on
each side all the way around. Here are some
comments on doing first layer corners
Completed first layer
Decide on a corner cubie you want to place in
position in the first layer. For example, the
red/blue/white corner. If that corner is already
in the first layer, then rotate it to the bottom
layer (and dont put in directly below the corner
that you want it at). Identify the target
location where you want to place it. In the
diagram above, it must go in the corner between
the red and blue faces. Rotate the red or blue
face so that the cubie in the target location is
also in the bottom layer, rotate the down face to
put the corner in position, and rotate the side
again to put the corner in the top layer. If you
get the corner in place but the orientation is
wrong, you have to try again. Try the same
process but from another direction. If the
corner you want to move has the white face on the
down side of the cube, put the corner directly
below where you want it to go in the top layer,
rotate the side containing that corner a quarter
turn so that the white face now is on either the
front, left or right side, rotate the down side,
rotate a side again to put the other white
facelets back in the top layer, and now the
corner cubie is ready to be moved as described
above.
10
The Middle Layer
To complete the middle layer, without messing up
the first layer, you need to learn two
algorithms
To solve the cube on your own, these are
algorithms you will want to memorize. The edges
of the middle layer (notice there are 4 such
edges) need to be placed to match the center
cubies for each side. Locate an edge you want to
move into the middle layer and position it as
shown in either of the figures below. In both
cases below, assume Fblue, Uwhite, Lred
Case 1 use D L D' L' D' F' D F
Case 2 use D' F' D F D L D' L'
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The Middle Layer
It might happen that a middle layer edge is
already in the middle layer but in the wrong
place or it might be in the right place but
oriented incorrectly as shown below.
To handle this situation, you need to move the
offending edge out of the middle layer and into
the last layer, and then move it back into the
middle layer the right way. To do that, just
use either of the previous algorithms to move the
piece back to the last layer.
Notice that as you use the middle layer
algorithms, there is an edge cubie that is
displaced each time the algorithm is applied.
The edge cubie that is in the target location is
moved to the last layer in both algorithms. So,
for example, to fix the problem in the diagram
above, we can move any other last layer edge into
the position currently held by the red/blue
cubie. Then the red/blue cubie will be in the
bottom layer and we can begin as already
described for solving the middle layer.
12
The Middle Layer
Now look at that! You are done with the middle
layer! You are 2/3 of the way through!!!

• Well, the last-layer has 4 steps and we need to
  learn 7 different algorithms if we are going to
  get those last cubies into place without messing
  up the work we have already done.
• These are last-layer steps
• First, flip the cube over so that the yellow
  side is the up side.
• Make a yellow cross.
• Permute the last layer corners
• Orient the last layer corners
• Permute the last layer edges

13
The Last Layer
Step 1 for the last layer Make the yellow
cross Make sure the cube is flipped over now, so
the yellow side (opposite white) is the up
side. If the first and middle layers are
complete, then there are only 4 possibilities for
the yellow facelets on the yellow side
Case 1 The yellow cross is complete, go to the
next step
Case 2 Turn the whole cube so that this is how
the yellow face looks and it is the up face. Do
the algorithm F U R U' R' F'
Case 3 Turn the whole cube so that the up face
looks exactly like this. Now do the algorithm
F R U R' U' F'
Case 4 Do either of the above algorithms. You
will get either case 2 or 3. Then do the
appropriate algorithm to get to case 1.
14
The Last Layer
Step 2 Swapping adjacent corners. Now inspect
the cube. See if all corner cubies in the last
(now the top) layer are in their proper
locations. It is ok if they are not oriented
correctly (their facelets may be facing the wrong
way). Also, note that the edges of the cross may
not line up with the corresponding center cubies
of adjacent faces. We fix that in step 4. Just
look to check if the corner cubies of the
last-layer are in the correct place (even if the
facelets are not oriented properly). Do this by
checking that the colors of the facelets besides
yellow match the colors of the faces making that
corner. There are two possibilities (1) they
are in the correct position or (2) two of them
are swapped out of place. You may need to rotate
the last-layer several times to determine which
case you have. If two are out of place, either
they are diagonally opposite to each other or the
two are adjacent corners. We will use an
algorithm for swapping adjacent last-layer
corners so if the corners you need to swap are
not adjacent, you have to do this algorithm twice.
15
The Last Layer
Step 2 Swapping adjacent corners. To swap
adjacent corners Hold the cube so that the two
corners that will be swapped are in the
front-top-right and back-top-right of the
cube. Use this algorithm L U' R' U L' U' R U2
Remember, if you need to swap diagonal corners
youll need to use this algorithm twice.
16
The Last Layer
Step 3 Orient the last layer corners Here you
can imagine we want to twist three of the corner
cubies so that the facelets are properly
oriented. There are two algorithms to learn
             
             
Case 1 twist three corner cubies clockwise Use
the algorithm R U R' U R U2 R' U2
Case 2 twist three corner cubies
counter-clockwise Use the algorithm R' U' R U'
R' U2 R U2
17
The Last Layer
Step 3 Orient the last layer corners
             
             
Case 1 twist three corner cubies clockwise Use
the algorithm R U R' U R U2 R' U2
Case 2 twist three corner cubies
counter-clockwise Use the algorithm R' U' R U'
R' U2 R U2
These are the only algorithms needed during this
step. It is important that the facelets of the
last layer are oriented exactly as shown in the
diagrams above before executing either
algorithm. These algorithms will move the
remaining facelets into place so that all of the
yellow facelets are then facing up on the up-side
of the cube.
18
The Last Layer
Step 3 To orient the last layer corners, you
may have one of the cases below. Rotate the cube
and apply either the case 1 or case 2 algorithm
to twist the facelets of three corner cubies so
that the result is either case 1 or case 2.
             
             
In this case, rotate the cube so it appears as
shown here. Then apply the case 1 (clockwise)
algorithm. The result is case 2.
Rotate the cube to this position and apply the
case 1 (clockwise) algorithm
Rotate the whole cube to this orientation and
apply the case 2 (counter-clockwise) algorithm
Rotate to this position and apply the case 2
(counter-clockwise) algorithm
Rotate to this position and apply the case 1
(clockwise) algorithm
19
The Last Layer
Step 4 Permuting the last layer edges
There are two algorithms to learn
             
             
Permute the edges clockwise. To do this use R2 U
F B' R2 F' B U R2
Permute the edges counter-clockwise. To do this
use R2 U' F B' R2 F' B U' R2
The other possible orientations of the last layer
edges are shown below.
If either of these cases occur, just use one of
the algorithms above which permute only three of
the four edges. After that, there will be one
edge in place and you will need to apply one of
the two algorithms above to complete the puzzle.
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Did you get it?
If you got it, then congratulations !! That is
quite an accomplishment! The solved state of
the cube is only 1 out of 43 quintillion possible
states!! Thats 1 out of 43,252,003,274,489,856,0
00 different possible states.