Me and Ernö Rubik in Budapest, Hungary, 2007 World Championships (25th anniversary of the original 1982 Rubik's Cube World Championships). This was the second world tournament Mr. Rubik attended, and announced this would be his last public appearance. Mr. Rubik is a Hungarian inventor, architect, professor of architecture, and best known for the invention of the Rubik's Cube (1974). The Rubik's Cube is one the top 10 most sold toys in the history of the world. Along with it, brought forth the study of its algorithms via Group Theory and computation. In 2010, Google used 35 CPU years worth of computation to prove that God's Number (the maximum minimum number of moves to solve any state of a 3x3x3 Rubik's Cube) is 20.

Contents

• Speedcubing: Objective
  

• Results; Competition, Performance
  

• Notation
  

• Beginner Method
  

• Intermediate Method
  

• Expert Method
  

• Beyond Expert Method
  

Speedcubing: Objective

\[ \text{Number of unique states for 3x3x3 cube:} \left(8!\right)\left(12!\right)\left(3^{7}\right)\left(2^{11}\right)\left(\frac{1}{2}\right) = 43,252,003,274,489,856,000 \]

Taking that number into perspective, at the rate of 1,000,000 arrangements per second, it would take well over a million years to run through them all. The actual complexity of this puzzle has brought forth mathematicians and computer scientists across the world to investigate the following question: what is the maximum number of moves to solve a Rubik's Cube from any given scrambled state? The answer to this question has been coined as “God's Number,” as one would suggest that God himself would use the minimum number of moves to solve it (using “God's Algorithm”). After \(\approx 30\) years of research on this problem, in 2010 an international team of researchers found that God's Number is equal to 20, requiring an equivalent of 35 years’ worth of computation time donated by Google. MIT researchers has shown that solving the Rubik's Cube is NP complete.

In 1982, the first Rubik's Cube World Championships took place in Budapest, Hungary, for humans competing to solve the \(3\)x\(3\)x\(3\) cube as fast as possible, resulting in a “speedcubing” community of competitors throughout the world. Different methods emerged: corners/edges (corners first, edges first), also layer-by-layer methods, and others. In speedcubing, the objective is to solve the Rubik's Cube in the fastest time possible, and to some degree, entailing minimizing the number of moves for a solve. Given how many possibilities there are for the Cube, there may be a threshold for how many algorithms a human can effectively use. For a set of algorithms known, \(\mathcal{S}\), average turn speed per move (efficiency), \(E=E\left(\mathcal{S}\right)\) (because some algorithms are faster to execute for human hands than others), and number of moves for a solve, \(N = N\left(\mathcal{S}\right)\),

\[ \begin{equation} \underset{\mathcal{S}}{\text{min }} t_\text{solve} = \left(\underset{\mathcal{S}}{\text{max }}E\left(\mathcal{S}\right) \right)\left(\underset{\mathcal{S}}{\text{min }}N\left(\mathcal{S}\right)\right)\label{eq:1} \end{equation} \]

Speedcubing then becomes an optimization problem of minimizing the number of moves (while also minimizing the execution) for a solve. The knowledge base of algorithms, \(\mathcal{S}\), to perform these optimizations, have been evolving over time, thanks to the community sharing newfound knowledge, in part due to computational tools available. My expert method has led me to compete internationally, and its set of algorithms \(\mathcal{S}\) can be found in the Expert Method section.

Results; Competition, Performance

Once the internet began to flourish in 2000, an underground speedcubing community formed online, and speedcubing tournaments started to emerge throughout the world. Thanks to the internet, I became a part of the international speedcubing community during this time of 2000-2007. The second world championships was the 2003 Rubik's Games World Championship which took place on August 23-24, 2003, in Tornoto, Canada. I competed in the 2003 Rubik's Games World Championship with the following times:

• 1st round (single): 24.19 seconds (18th place)

• Semi-final round (single): 23.96 seconds (23rd place)

• Final round: Took the top 8 from semi-final round

The third Rubik's world championship took place November 5-6, 2005, in Orlando, Florida at the Disney Pop Century Resort, Disney World. I competed in the 2005 Rubik's World Championships with the following times:

• 1st round (single): 18.64 seconds (26th place)

• Semi-final round (average): 16.29 seconds (5th place)

• Final round (average): 18.71 seconds (12th place)

The fourth Rubik's world championship took place October 5-7, 2007, in Budapest, Hungary. This was the 25th anniversary of the first Rubik's world championship. I traveled to Budapest right after a Physics exam in college. I was sponsored by the Department of Physics at the University of Arizona. I ended up performing my best official solving average of 15.62 seconds and single time of 14.18 seconds. I decided to retire (at least for a while) from competing in Rubik's tournaments seriously. I competed in the 2007 Rubik's World Championships with the following times:

• 1st round (average): 16.15 seconds (27th place)

• Semi-final round (average): 15.62 seconds (20th place)

• Final round (average): Took the top 16 from the semi-final round

Per the objective \(\eqref{eq:1}\), below one can see I attempted to minimize \(t_{\text{solve}}\) over time, as my minimum was converging to something close to \(\approx 14-15\) seconds in an official setting.

My biggest highlights,

• Fastest official average (single) in 2007 World Championships: 15.62 seconds (14.18 seconds). (2007)

• “24 hour marathon” world record holder in February 2006; 3141.5 \(3\)x\(3\)x\(3\) solves, thanks to a team of friends scrambling for me at Caltech. The previous record was 2,000 solves by Jess Bonde from Denmark. I solved 2,000 at the 12 hour mark. I held the record until June 2006 when Zbigniew Zborowski broke my record. (2006)

• At the 2005 Rubik's World Championships, I ranked fastest from USA in semi-final round, and top 12 in the final round; Orlando, Florida, Disneyworld. (2005)

• Almost became USA Champion at the 2004 USA Rubik's Championships. (2004)

As for media coverage and public appearances,

• Arizona, USA: Arizona Daily Star Newspaper: Interviewed twice for articles (2003 and 2005).

• Tucson, AZ, USA; Tucson Citizen: Interviewed once for article (2004).

• Tucson, AZ, USA; Arizona Wildcat: Interviewed twice by the University of Arizona's newspaper (2005 and 2007).

• Tucson, AZ, USA; Channel 4 News: News coverage about me and the 2007 Rubik's World Championships results (2007).

• Tucson, AZ, USA; Channel 13 News: Publicly seeking sponsors for the 2003 Rubik's “Games” World Championships (2003).

• Phoenix, AZ, USA; International Conference on Thinking XI (2003).

• Phoenix, AZ, USA; Diamondbacks baseball game (2003): 7th inning stretch, jumbotron screen

• Phoenix, AZ, USA; Phoenix Suns vs Los Angeles Lakers Basketball Game: commercial on the Jumbo Screen (2004)

• Tucson, AZ, USA; University High School Talent Show; received a standing ovation in 2004 and won the first money prize for solving the Rubik's Cube behind my back (blindfolded) (2004 and 2005)

• Tucson, AZ, USA; University of Arizona “Physics Phun Night” (2007 and 2008)

Notation

Color Scheme

Blue-opposite-green, white-opposite-yellow, red-opposite-orange will be the color scheme used. This is a very common color scheme If you have a different color scheme, it's fine; the algorithms and concepts still apply.

The Pieces, and Miscellaneous Terminology

All edges remain edges, all corners remain corners, and the centers remain centers. There are 8 corners, 12 edges, and 6 centers. For each piece, or "cubie," there are stickers; an edge has two stickers, a corner has three, and a center has 1. These stickers are called "facets." A layer is a row of cubies; for the Rubik's Cube shown above, there are three layers: a top layer, middle layer, and bottom layer.

The Faces

There are letters corresponding to each face of the Cube: R is for right face; L is for left face; U is for up (or top) face; D is for down (or bottom) face; F is for front face; B is for back face. The F face should be facing you, and the U face should be facing the ceiling, at all times when applying moves and algorithms. Note: the faces are determined by the center piece color. The reason for this is because center pieces never move, so it allows us to use consistent notation. For the defined color scheme, F face is white, B face is yellow, U face is blue, D face is green, R face is red, and L face is orange. For no rotations of the cube, this will remain consistent.

Face Turns
For notational purposes, the letters themselves, when explaining a move, mean to move that face clockwise. An apostrophe (" ’ ") after the letter means to move that face counterclockwise, and a "2" after the letter means to move that face twice (this is a double turn). Note: for a double turn, the direction doesn't matter. Envision a clock at the center of each face to determine the clockwise or counterclockwise direction of each face. To simplify this, see the following:
Face		Clockwise		Counter- Clockwise		Double Turn (any direction)
Right		R		R’		R2 or R2’
Left		L		L’		L2 or L2’
Up (top)		U		U’		U2 or U2’
Down (bottom)		D		D’		D2 or D2’
Front		F		F’		F2 or F2’
Back		B		B’		B2 or B2’
Further Examples
Let us consider the following as the hypothetical initial state of the Cube: Below is a table of the move, and "aftereffect" of what that move does to the initial state, to further exemplify the move's effect.

Move	Aftereffect		Move	Aftereffect		Move	Aftereffect
R			R’			R2 or R2’
L			L’			L2 or L2’
U			U’			U2 or U2’
D			D’			D2 or D2’
F			F’			F2 or F2’
B			B’			B2 or B2’

Slice Turns

Sometimes it's useful to make slice turns. There are three general directions for slice moves: M (middle), E (equator), and S (side). The M, E, and S are subject to a " ’ " and a "2" at the end of the letter. Visually, the moves are as follows:

Further Examples

Let us consider the following as the hypothetical initial state of the Cube: Below is a table of the move, and "aftereffect" of what that move does to the initial state, to further exemplify the move's effect.

Move	Aftereffect		Move	Aftereffect		Move	Aftereffect
M			M’			M2 or M2’
E			E’			E2 or E2’
S			S’			S2 or S2’

Rotations
Sometimes it's useful to make a rotation. There are three general directions for rotations: about the x-axis, y-axis, and z-axis, referring to the 3 dimensional cartesian coordinate system. The x,y, and z are subject to a " ’ " and a "2" at the end of the letter. To determine a clockwise or counterclockwise direction, envision a clock on the cube observing from the positive end of that axis. To simplify this, see the following:
Rotation	Clockwise	Counter- clockwise	Double Turn (any direction)
About x-axis	x	x’	x2 or x2’
About y-axis	y	y’	y2 or y2’
About z-axis	z	z’	z2 or z2’
Further Examples
Let us consider the following as the hypothetical initial state of the Cube: Below is a table of the move, and "aftereffect" of what that move does to the initial state, to further exemplify the move's effect.

Move	Aftereffect		Move	Aftereffect		Move	Aftereffect
x			x’			x2 or x2’
y			y’			y2 or y2’
z			z’			z2 or z2’

Double Layer Turns
For double layer turns, use of the lowercase letter of a face (L,R,U,D,F,B). A double layer turn is similar to a face turn, but instead of moving a single layer, a double layer is turned. These are subject to " ’ " and "2" at the end of each letter.
Two Layers		Clockwise		Counter- Clockwise		Double Turn (any direction)
Right + Middle		r = (R M’)		r’ = (R’ M)		r2 = (R2 M2’) or r2’ = (R2’ M2)
Left + Middle		l = (L M)		l’ = (L’ M’)		l2 = (L2 M2) or l2’ = (L2’ M2’)
Up (top) + Equator		u = (U E’)		u’ = (U’ E)		u2 = (U2 E2’) or u2’ = (U2’ E2)
Down (bottom) + Equator		d = (D E)		d’ = (D’ E’)		d2 = (D2 E2) or d2’ = (D2’ E2’)
Front + Slice		f = (F S)		f’ = (F’ S’)		f2 = (F2 S2) or f2’ = (F2’ S2’)
Back + Slice		b = (B S’)		b’ = (B’ S)		b2 = (B2 S2’) or b2’ = (B2’ S2)
Further Examples
Let us consider the following as the hypothetical initial state of the Cube: Below is a table of the move, and "aftereffect" of what that move does to the initial state, to further exemplify the move's effect.

Move	Aftereffect		Move	Aftereffect		Move	Aftereffect
r			r’			r2
l			l’			l2
u			u’			u2
d			d’			d2
f			f’			f2
b			b’			b2

Facet Notation

There is a facet notation (developed by mathematician David Singmaster) that is sometimes used to distinguish all facet positions. If we keep the same color scheme, we can represent each facet using the following notation: The first letter represents the actual face the facet is on. For an edge, the second letter is the other facet for the edge; for a corner, the remaining two letters represent the faces as they appear going in a clockwise direction.

Beginner Method

Step 1: Cross

Goal

The D center will always be there, so the center of the cross is always done. Four edges are to be placed correctly to form a cross. The desired destination for each cross edge will be denoted as the "empty slot," highlighted in a light gray color, to indicate where the cross color will end up after you perform an algorithm for that cross edge. The empty slot should always be on the bottom layer, facing you (so the empty slot for the cross color is in the DF position):

Cases for the Cross on Bottom
Case	Case	Algorithm		Case	Case	Algorithm
1.1		F2		1.2		F D R’
1.3		D R’		1.4		F
1.5		F’ D R’		1.6		Nothing, since this is what you want. Repeat setup/insertion for the next cross edge

After repeating the above process for all four cross edges, you should have the following at this point:

Cases for Matching the Cross Edges with Adjacent Centers
Case	Case	Algorithm
1.7	Swap two adjacent cross edges; this swaps the front bottom edge with the right bottom edge	R2 U F2 U’ R2 or R’ D R D’ R’
1.8	Swap two opposite cross edges; this swaps the front bottom edge with the back bottom edge	M2 U2 M2
1.9	Solved already	Nothing, since this is what you want.

Step 2: First Layer

Goal

Cases for the First Layer Corners
Case	Case	Algorithm		Case	Case	Algorithm
2.1		U’ F’ U F or R U R’		2.2		U R U’ R’ or F’ U’ F
2.3		U R U2 R’ then case 2.2		2.4		R U’ R’ then case 2.2
2.5		F’ U F then case 2.1		2.6		Nothing, since this is what you want. Repeat setup/insertion for the next first layer corner

By now you should have step 2 completed, after inserting all four corners properly. If you have a corner placed in the wrong location (because that's where the corner was, or because you put it there accidentally), have it be in the front right bottom (FRD/RDF/DFR) corner location and perform R U’ R’ to move the improperly placed corner in the U layer, now enabling you to insert the corner to its proper location.

Step 3: Second Layer

Goal

Now that we've finished the first layer, we can focus on the second layer edges to complete the second layer. For inserting second layer edges, I will indicate the the empty slot (which is the desired destination for the second layer edge you are trying to insert properly) by making it a light gray color, which will be either on the front right (FR/RF) or front left (FL/LF) edge location.

Cases for the Second Layer Edges
Case	Case	Algorithm		Case	Case	Algorithm
3.1		U R U’ R’ U’ F’ U F		3.2		y U’ L’ U L U F U’ F’ y’
3.3		U R U’ R’ U’ F’ U F U2 then case 3.1 or (R U2 R’ U)x2 F’ U’ F		3.4		Nothing, since this is what you want. Repeat setup/insertion for the next second layer edge

Step 4: Last Layer Corners

Goal

Now that we've finished the first two layers, we can focus on the third layer corners to partially complete the last layer. For this step, we will repeatedly a) set up our Cube to one of the following cases using U moves and/or y rotations, b) perform the algorithm, and then repeat a and b until the corners for the last layer are solved. In the setups below, the F face corresponds to the white center as usual.

Cases for the Last Layer Corners
Case	Case	Algorithm		Case	Case	Algorithm
4.1		R’ U’ R U’ R’ U2 R U2, repeat setup (if necessary)		4.2		Same as Algorithm 4.1, then repeat setup (if necessary)
4.3		Same as Algorithm 4.1, repeat setup (if necessary)		4.4		Nothing, since this is what you want.

After orienting/flipping/twisting the corners correctly, we now have to permute/place the corners in such a way that eaach corner is in its solved state. Just because the corners are twisted properly doesn't necessarily mean they are in their proper place. There are three possibilities that may arise to complete the last layer corners:

Cases for Permuting the Last Layer Corners While Preserving Corner Orientation
Case	Case	Algorithm
4.5	Swap a pair of corners: swapping the two top corners on the right side. Viewed from above: Example where white/orange/green corner and yellow/orange/green corner need to be swapped.	U L F’ L F2 R’ F R F2 L2
4.6	Swap two pairs of corners: swapping the two top corners on the right side, and swapping the two top corners on the left side. Viewed from above: Example where white/orange/green corner and yellow/orange/green corner need to be swapped, and the white/red/green and yellow/red/green corner need to be swapped:	U2, Algorithm 4.5, U2 then case 4.5 or U R2 D’ L F’ L’ D R2 U B U B’
4.7	Solved already	Nothing, since this is what you want.

Step 5: Last Layer Edges

Goal

Now that we've finished the last layer corners, we can focus on the last layer edges to complete the Cube. We will need to focus on the actual positions of the edges first, before flipping them correctly. Make only y rotations to find an edge that is inserted in its correct place (note: we aren't allowed U moves anymore because the last layer corners are now in place, so we are in a way "locking" them into place). Once you find an edge that is in its correct place, then make that face your F face, regardless of whether or not that last layer edge on the F face is flipped correctly or not. If you cannot find an edge that is located in its proper place, then you can make any face your F face. Below is an algorithm which you will have to repeat until all edges are located in their proper place.

Cases for Permuting the Last Layer Corners
Case	Case	Algorithm
5.1	An edge is in its proper place: make that face the F face when performing algorithm. Example with white/green edge: or	M’ U’ M U2 M’ U’ M then find another edge that is located properly, do y rotations to make it your new F face (until all edges are solved)
5.2	None of the last layer edges are placed properly: make any face the F face when performing algorithm.	Algorithm 5.1

After placing the edges correctly, all that is left to do is flip the last layer edges. There are only three cases that may arise in this step. Match up the Cube to one of the following setups and perform the algorithm.

Cases for Orienting/Flipping the Last Layer Edges
Case	Case	Algorithm
5.3	The "H" pattern. The only two edges needed to be flipped are the F and B edges on the top layer. Example with white/green and yellow/green edges:	(M’ U’)x2 M’ U2 (M U’)x2 M U2
5.4	The only two edges needed to be flipped are the F and R edges on the top layer. Example with white/green and orange/green edges:	R B, Algorithm 5.3, B’ R’
5.5	None of the edges need to be flipped. Example:	Nothing, since this is what you want.

Intermediate Method

Step 1: Cross

Goal

The D center will always be there, so the center of the cross is always done. You will have to place four edges correctly to form a cross. The cross edges should also match up with the adjacent center pieces for the L, R, F, B faces. As an intermediate, this should be done in one step using intuition. From here onward my F face will be the white side, R will be orange, L will be red, B will be yellow (D will still be blue, and U will still be green).

Step 2: First Layer

Goal

Now that we've finished the cross, with the cross edges also matching up with the adjacent center pieces, we can focus on the first layer corners to complete the first (bottom) layer. For inserting first layer corners, I will place the empty slot (which is the desired destination for the first layer corner you are trying to insert properly) on the front right bottom (FRD/RDF/DFR) corner location.

Cases for the First Layer Corners
Case #	Case	Algorithm		Case #	Case	Algorithm
2.1		U’ F’ U F or R U R’		2.2		U R U’ R’ or F’ U’ F
2.3		U R U2 R’ then case 2.2		2.4		R U’ R’ then case 2.2
2.5		F’ U F then case 2.1		2.6		Nothing, since this is what you want. Repeat setup/insertion for the next first layer corner

By now you should have step 2 completed, after inserting all four corners properly. If you have a corner placed in the wrong location (because that's where the corner was, or because you put it there accidentally), have it be in the front right bottom (FRD/RDF/DFR) corner location and perform R U’ R’ to move the improperly placed corner in the U layer, now enabling you to insert the corner to its proper location.

Step 3: Second Layer

Goal

Cases for the Second Layer Edges
Case #	Case	Algorithm		Case #	Case	Algorithm
3.1		U R U’ R’ U’ F’ U F		3.2		y U’ L’ U L U F U’ F’ y’
3.3		U R U’ R’ U’ F’ U F U2 then case 3.1 or (R U2 R’ U)x2 F’ U’ F		3.4		Nothing, since this is what you want. Repeat setup/insertion for the next second layer edge

Step 4: Last Layer Oriention

Goal

Cases for Last Layer Orientation
Case #	Case	Algorithm		Case #	Case	Algorithm
4.1		F U R U’ R’ then cases 4.2 - 4.3		4.2		Algorithm 4.1 then cases 4.4 - 4.10
4.3		F R U R’ U’ F’ then cases 4.4 - 4.10		4.4		R U R’ U R U’ R’ U R U2 R’
4.5		R U2 R2 U’ R2 U’ R2 U2 R		4.6		R’ F’ L F R F’ L’ F
4.7		L2 D’ L U2 L’ D L U2 L		4.8		R’ F’ L’ F R F’ L F
4.9		R’ U2 R U R’ U R		4.10		L U2 L’ U’ L U’ L’ (reflection of 4.9)

After orienting the last layer, we now have to permute/place the last layer.

Step 5: Permuting Last Layer

Goal

Cases for Last Layer Corner Permutation Only, Ignoring Edge Permutation
Case #	Case	Algorithm		Case #	Case	Algorithm
5.1		R2 B2 R F R’ B2 R F’ R (reflection of 5.2)		5.2		R’ F R’ B2 R F’ R’ B2 R2 (reflection of 5.1)
5.3		Algorithm 5.1, U’ y’ then case 5.2 or F R U’ R’ U’ R U R’ F’ R U R’ U’ R’ F R F’		5.4		U’ y2 then case 5.2
5.5	Corners are already permuted:	Nothing, since this is what you want.

Cases for Last Layer Edge Permutation While Preserving Corner Permutation
Case #	Case	Algorithm		Case #	Case	Algorithm
5.5		R2 U’ F B’ R2 F’ B U’ R2 or R2 U’ y R’ L F2 L’ R y’ U’ R2 (reflection of 5.6)		5.6		R2 U F B’ R2 F’ B U R2 or R2 U y R’ L F2 L’ R y’ U R2 (reflection of 5.5)
5.7		Algorithm 5.5, y’ then case 5.5 or U’ L U L’ U L’ U’ L’ U L U’ L’ U’ L2 U L U’		5.8		M2 U’ M2 U2 M2 U’ M2
5.9	Edges are permuted	Nothing, since this is what you want.

You have now solved the Cube! To recap, we did five steps: the cross, the first layer, the second layer, the last layer orientation, and then last layer permutation. For the last layer orientation, we oriented the last layer edges first (if they weren't already), then the corners were oriented. For the last layer permutation, we permuted the corners first, then the edges. The number of algorithms you need to know for this is broken down in the following:

Expert Method

The D center will always be there, so the center of the cross is always done. You will have to place four edges correctly to form a cross. The cross edges should also match up with the adjacent center pieces for the L, R, F, B faces. As an expert, this should be done in one step using intuition, using no more than 9 moves (it has been proven/shown that 9 moves is the max for this step, and 99.5% of the time 8 the maximum number of moves). From here onward my F face will be the white side, R will be orange, L will be red, B will be yellow (D will still be blue, and U will still be green).

Step 2: First Two Layers

Goal

Now that we've finished the cross, with the cross edges also matching up with the adjacent center pieces, we can focus on the first two layers. We will form the first two layers by combining the first layer corners with the second layer edges, by forming corner/edge (C/E) pairs. For inserting C/E pairs, I will place the empty slot (which is the desired destination for the C/E pair) on the front right (FR/RF) location.

Cases for the First Two Layers Corner/Edge Pair
Case #	Case	Algorithm		Case #	Case	Algorithm
2.1		U F’ U’ F U2 F’ U F’		2.2		U’ R U R’ U2 R U’ R’
2.3		U R U’ R’ U’ F’ U F or y’ (L’ U’)x3 (L U)x3 or D U’ F’ U F D’ (if not worrid about preserving FL/LF C/E pair)		2.4		y U’ L’ U L U F U’ F’ or (R U)x3 (R’ U’)x3 or D U R U’ R D’ (if not worried about preserving FL/LF C/E pair)
2.5		U (R U’ R’ U)x2 R U’ R’ or (R’ F R F’)x3		2.6		R U’ R’ U F’ U F or F’ R U’ R’ U R’ F R (if not worried about preserving FL/LF C/E pair)
2.7		R U R’ U’ then case 2.26		2.8		R U R’ U’ then case 2.30
2.9		R U R’ U’ then case 2.24 or R U’ R’ F’ L’ U2 L F		2.10		y L’ U’ L U then case 2.23 or y L’ U L F R U2 R’ F’
2.11		y L’ U L U’ L’ U L or y L U L2 F’ L’ F		2.12		R U’ R’ U R U’ R’ or R U’ R2 F R F’
2.13		(U’ R U R’)x2		2.14		y (U L’ U’ L)x2
2.15		y U (L’ U L) U2’ (L’ U L)		2.16		U’ (R U’ R’) U2’ (R U’ R’)
2.17		U y (L’ U’ L) U’ y’ (R U R’)		2.18		U’ (R U R’) U F’ U F
2.19		U’ F’ U F		2.20		U R U’ R’
2.21		U R U’ R’ then case 2.13 or R U2 R2 U R’ (if not worried about preserving BR/RB C/E pair)		2.22		y U’ L’ U L then case 2.14 or y L U2 L2 U’ L (if not worried about preserving BL/LB C/E pair)
2.23		F’ U F U2 R U R’		2.24		R U’ R’ U2 F’ U’ F
2.25		y U L’ U L U’ L’ U’ L		2.26		U’ R U’ R’ U R U R’
2.27		U R U’ R’ U’ then caes 2.12		2.28		y U’ L’ U L U then case 2.11
2.29		y L’ U2 L U L’ U L		2.30		R U2 R’ U’ R U’ R’
2.31		U2 R U’ R’ U’ R U R’		2.32		y U2 L’ U L U L’ U’ L
2.33		U’ R U2 R’ U2 R U’ R’		2.34		y U L’ U2 L U2 L’ U L’
2.35		F’ U’ F		2.36		R U R’
2.37		U’ y L’ U2 L U’ L’ U L		2.38		U R U2 R’ U R U’ R’
2.39		R U’ R’ U2’ R U R’		2.40		y U2 L’ U’ L U’ L’ U L
2.41		(R U2 R’ U)x2 F’ U’ F		2.42		Nothing, since this is what you want. Repeat setup/insertion for the next corner/edge pair (if necessary).

Step 3: Last Layer Orientation

Goal

Cases for Last Layer Orientation
Case #	Probability	Case	Algorithm		Case #	Probability	Case	Algorithm
3.1	1/108		R U2 R2 F R F’ U2 R’ F R F’		3.2	1/54		F R’ F’ R U R2 B’ R’ B U’ R’
3.3	1/27		R L2 B’ L B’ R’ U2 R B’ L R’		3.4	1/54		L’ U’ L U’ L F’ L’ F U2 L F’ L’ F
3.5	1/54		L F R U2 R’ U2 R U2 R’ F’ L’		3.6	1/54		R’ U2 F R U R’ U’ F2 U2 F R
3.7	1/216		(M U’)x4 (M’ U’)x4		3.8	1/108		R U2 R2 U’ R U’ R’ U2 y’ R B R’
3.9	1/108		R B R’ U L U’ L’ U L U’ L’ R B’ R’		3.10	1/54		R U’ y L’ U L U’ L’ U L F
3.11	1/54		R’ U’ R U’ R’ U y L’ U L U’ L’ U L F		3.12	1/27		R B L’ B L B’ L’ B L B2 R’
3.13	1/27		R y’ R’ U2 R U2 R B2 R’ B		3.14	1/27		F (R U R’ U’)x2 F’
3.15	1/27		R B L’ B L B2 R’		3.16	1/27		L2 R B’ L B’ L’ B2 L B’ L R’
3.17	1/27		L’ U L y’ L F’ L’ U L F L’		3.18	1/27		L B2 R B R’ B L
3.19	1/27		L F’ L’ U’ L F L’ y L’ U L		3.20	1/27		R B R’ L U L’ U’ R B’ R’
3.21	1/27		R U’ R’ U2 R U y R U’ R’ U’ F’		3.22	1/27		B L B’ R B L2 B L B2 R’
3.23	1/27		R U B’ U’ R’ U R B R’		3.24	1/54		F R U’ R’ U’ R U R’ F’
3.25	1/27		L d R U’ R’ F’		3.26	1/54		R U2 R2 F R F’ R U2 R’
3.27	1/54		F R U R’ U’ F’		3.28	1/54		R U R’ U’ R’ F R F’
3.29	1/27		L F’ L’ U’ L U F U’ L’		3.30	1/54		L U L’ U’ y L’ B’ R B R’ L
3.31	1/54		R U R B’ R’ B U’ R’		3.32	1/27		L U L’ U L U’ L’ U’ L’ B L B’
3.33	1/108		L U2 L’ U’ L U L’ U’ L U’ L’		3.34	1/54		R U2 R2 U’ R2 U’ R2 U2 R
3.35	1/54		L2 D’ L U2 L’ D L U2 L		3.36	1/54		R’ F’ L F R F’ L’ F
3.37	1/54		R F’ L’ F R F’ L F		3.38	1/27		L U2 L’ U’ (L U’ L’)
3.39	1/54		L’ B’ (3.40) B L		3.40	1/108		(M U’)x2 M U2 (M’ U’)x2 M’ U2
3.41	1/216		Nothing; desired state

Step 4: Last Layer Permutation

Goal

Cases for Last Layer Permutation
Case #	Probability	Case	Algorithm		Case #	Probability	Case	Algorithm
4.1	1/9		(L2 U) (L U L’) (U’ L’) (U’ L’ U L’)		4.2	1/9		L2 B2 (L’ F’ L) B2 (L’ F L’)
4.3	1/36		U’ (L U L’ U) (L’ U’) L’ U (L U’ L’ U’) L2 U L U’		4.4	1/72		M2 U’ M2 U2 M2 U’ M2
4.5	1/36		F R’ F’ L F R F’ L2 B’ R B L B’ R’ B		4.6	1/18		R U R’ U’ R’ F R2 U’ R’ U’ R U R’ F’
4.7	1/18		R’ U R’ U’ y z U’ R U’ R’ U2 y’ R’ U’ R U R		4.8	1/18		R’ U R U’ R2 y’ R’ U’ R U y x R U R’ U’ R2 B’
4.9	1/9		R’ U2 R U2 R’ F R U R’ U’ R’ F’ R2 U’		4.10	1/9		B2 L U L’ B2 l x’ U’ R U R2
4.11	1/18		F R U’ R’ U’ R U R’ F’ R U R’ U’ R’ F R F’		4.12	2/9		(L U L’) B2 D’ R U’ R’ U R’ u R2
4.13	1/36		(L U’ R U2 L’ U R’)x2 U		4.14	1/72		Nothing; desired state

The number of algorithms you need to know for this is broken down in the following:

Beyond Expert Method