Hold the cube with one side parallel to the ground and one side facing you. The six sides of the cube are called faces and are denoted by six letters:

F (Front) - the side facing you.

U (Up) - the side facing upwards.

R (Right) - the side facing to the right.

B (Back) - the side facing away from you.

L (Left) - the side facing to the left.

D (Down) - the side facing downwards.

A Rubik's Cube is made up of six center pieces, which cannot change their position relative to each other, eight corner pieces, and twelve edge pieces. Each face of the cube corresponds to an outer layer: The U layer, for example, consists of the four edge pieces, four corner pieces, and one center piece that are currently visible on the U face of the cube. There is a distinction between the pieces that are currently in the U layer, and the actual U layer pieces, i.e. the pieces that will be in the U layer when the cube is solved without changing the positions of the centers (i.e. without rotating the cube).

There are exactly three different turns that can be applied to each of the six outer layers. U, F, R, B, L and D denote a 90 degree turn in clockwise direction of the corresponding layer. The term 'clockwise' refers to the direction to turn the layer, if you are looking directly at the corresponding face. U', F', R', B', L' and D' denote a 90 degree counterclockwise turn of the corresponding layer, and U2, F2, R2, B2, L2 and D2 denote a 180 degree turn. U' is pronounced U prime or U inverse.

Each edge piece is located in two outer layers at the same time. Hence an edge piece can be described to be in the UF position, if it is currently located in the U and F layers. The edge piece that will be in the UF position when the cube is solved without changing the positions of the centers is called the UF edge piece. The four edge pieces that will be in the U layer when the cube is solved are called U layer edges. A corner piece is located in three outer layers at the same time and can be said to be in the UFR position, if it is in the U, F and R layers. It can be said to be the UFR corner piece, if it will be in the UFR position when the cube is solved without changing the positions of the centers. The four corner pieces that will be in the U layer when the cube is solved are called U layer corners, and the two corner pieces that will be in the U and R layers when the cube is solved are called UR layer corners.

The inner layers are called slices and contain 4 edge pieces and 4 center pieces. The M slice (Middle layer) is the layer between the L and R layers, the E slice (Equatorial layer) is the layer between the U and D layers, and the S slice (Standing layer) is the layer between the F and B layers. An M turn is a 90 degree turn of the M slice in the same direction as an L turn. An E turn is a 90 degree turn of the E slice in the same direction as a D turn. An S turn is a 90 degree turn of the S slice in the same direction as an F turn. Double-layer turns move two adjacent layers at once and are denoted by u, f, r, b, l and d. A u turn is a 90 degree turn of the U and E layers in the same direction as a U turn. Of course there are also M', M2, u', u2 turns that move the layers in the opposite direction or by 180 degrees.

A cube position is said to be in the <L, R, U> group, if it can be solved using exclusively L, R and U turns. It is in the <R, U> group, if the position can be solved using exclusively R and U turns. Cube positions in the <L, R, U> group are called 3-gen (which stands for 3 generators) and cube positions in the <R, U> group are called 2-gen.

F (Front) - the side facing you.

U (Up) - the side facing upwards.

R (Right) - the side facing to the right.

B (Back) - the side facing away from you.

L (Left) - the side facing to the left.

D (Down) - the side facing downwards.

A Rubik's Cube is made up of six center pieces, which cannot change their position relative to each other, eight corner pieces, and twelve edge pieces. Each face of the cube corresponds to an outer layer: The U layer, for example, consists of the four edge pieces, four corner pieces, and one center piece that are currently visible on the U face of the cube. There is a distinction between the pieces that are currently in the U layer, and the actual U layer pieces, i.e. the pieces that will be in the U layer when the cube is solved without changing the positions of the centers (i.e. without rotating the cube).

There are exactly three different turns that can be applied to each of the six outer layers. U, F, R, B, L and D denote a 90 degree turn in clockwise direction of the corresponding layer. The term 'clockwise' refers to the direction to turn the layer, if you are looking directly at the corresponding face. U', F', R', B', L' and D' denote a 90 degree counterclockwise turn of the corresponding layer, and U2, F2, R2, B2, L2 and D2 denote a 180 degree turn. U' is pronounced U prime or U inverse.

Each edge piece is located in two outer layers at the same time. Hence an edge piece can be described to be in the UF position, if it is currently located in the U and F layers. The edge piece that will be in the UF position when the cube is solved without changing the positions of the centers is called the UF edge piece. The four edge pieces that will be in the U layer when the cube is solved are called U layer edges. A corner piece is located in three outer layers at the same time and can be said to be in the UFR position, if it is in the U, F and R layers. It can be said to be the UFR corner piece, if it will be in the UFR position when the cube is solved without changing the positions of the centers. The four corner pieces that will be in the U layer when the cube is solved are called U layer corners, and the two corner pieces that will be in the U and R layers when the cube is solved are called UR layer corners.

The inner layers are called slices and contain 4 edge pieces and 4 center pieces. The M slice (Middle layer) is the layer between the L and R layers, the E slice (Equatorial layer) is the layer between the U and D layers, and the S slice (Standing layer) is the layer between the F and B layers. An M turn is a 90 degree turn of the M slice in the same direction as an L turn. An E turn is a 90 degree turn of the E slice in the same direction as a D turn. An S turn is a 90 degree turn of the S slice in the same direction as an F turn. Double-layer turns move two adjacent layers at once and are denoted by u, f, r, b, l and d. A u turn is a 90 degree turn of the U and E layers in the same direction as a U turn. Of course there are also M', M2, u', u2 turns that move the layers in the opposite direction or by 180 degrees.

A cube position is said to be in the <L, R, U> group, if it can be solved using exclusively L, R and U turns. It is in the <R, U> group, if the position can be solved using exclusively R and U turns. Cube positions in the <L, R, U> group are called 3-gen (which stands for 3 generators) and cube positions in the <R, U> group are called 2-gen.

The current state of the cube is called cube position or cube permutation. The set of cube positions that contains the solved state as well as all the cube positions that can be reached by applying a sequence of U, F, R, B, L and D turns to the solved cube are called legal positions.

An edge is said to be a good edge or a (correctly) oriented edge, if it can be solved using exclusively L, R, U and D turns. Otherwise the edge is called a bad edge or an unoriented edge. Changing the positions of the centers (e.g. by rotating the cube or by turning the slices) can change the orientation of an edge. To find all the bad edges of the current cube position, look at each of the six faces of the cube one by one (but do not alter your grip to avoid rotating the cube).

If you see a sticker on the U or D faces that belongs to an edge and that has the same colour as the L or R center pieces, then this edge is a bad edge.

If you see a sticker on the L or R faces that belongs to an edge and that has the same colour as the U or D center pieces, then this edge is a bad edge.

If you see a sticker on the F or B faces that belongs to an edge of the E slice and that has the same colour as the L or R center pieces, then this edge is a bad edge.

If you see a sticker on the F or B faces that belongs to an edge of the U or D layer and that has the same colour as the U or D center pieces, then this edge is a bad edge.

U, D, L and R turns do not change the orientation of an edge piece. An F turn changes the orientation of all the edges on the F layer and a B turn changes the orientation of all the edges on the B layer. If exactly one edge on the F layer is oriented, then exactly three edges will be oriented after an F turn, and the the number of oriented edges is changed by two. When exactly two edges on the F layer are oriented, an F turn changes the number of oriented edges by two, and if all or none edges on the F layer are oriented, the number of oriented edges does not change. Since all the edges of a solved cube are oriented and since slice turns and double-layer turns are just combinations of outer layer turns, all legal cube positions have an even number of oriented edges.

A corner is said to be an oriented corner or a corner of orientation 0, if it is located on the D layer and the sticker facing down has U or D color, or if it is located in the U layer and the sticker facing up has U or D color. A corner has orientation 1, if twisting it counterclockwise would leave it oriented correctly, and it has orientation 2, if twisting it clockwise would orient it.

Given two positive numbers a and n, a modulo n is the remainder of the division of a by n. For example, 5 modulo 3 is equal to 2. We write 5 = 2 mod 3. Now let a be the sum of the orientations of the eight corners. The remainder of the division of a by 3 is a number q between 0 and 2. We can write: a = q mod 3. If we now twist a single corner, the remainder of the division changes and we get a new remainder q'. It holds that q' = q + 1 mod 3 if the corner is twisted clockwise, and q' = q + 2 mod 3 if it is twisted counterclockwise.

Every outer layer turn twists two corners clockwise and two corners counterclockwise. So a single outer layer turn changes the remainder as follows: q' = q + 1 + 1 + 2 + 2 mod 3 = q + 6 mod 3 = q + 0 = q. So the remainder does not change at all. This means that all legal cube positions have the same remainder q. Since q = 0 if the cube is solved, we now know that the remainder of a legal cube position is always zero, i.e. the sum of the orientations of the eight corners is divisible by 3. This means that if you know the orientation of seven corner pieces of a legal cube position, the orientation of the eigth corner piece is uniquely determined.

An edge is said to be a good edge or a (correctly) oriented edge, if it can be solved using exclusively L, R, U and D turns. Otherwise the edge is called a bad edge or an unoriented edge. Changing the positions of the centers (e.g. by rotating the cube or by turning the slices) can change the orientation of an edge. To find all the bad edges of the current cube position, look at each of the six faces of the cube one by one (but do not alter your grip to avoid rotating the cube).

If you see a sticker on the U or D faces that belongs to an edge and that has the same colour as the L or R center pieces, then this edge is a bad edge.

If you see a sticker on the L or R faces that belongs to an edge and that has the same colour as the U or D center pieces, then this edge is a bad edge.

If you see a sticker on the F or B faces that belongs to an edge of the E slice and that has the same colour as the L or R center pieces, then this edge is a bad edge.

If you see a sticker on the F or B faces that belongs to an edge of the U or D layer and that has the same colour as the U or D center pieces, then this edge is a bad edge.

U, D, L and R turns do not change the orientation of an edge piece. An F turn changes the orientation of all the edges on the F layer and a B turn changes the orientation of all the edges on the B layer. If exactly one edge on the F layer is oriented, then exactly three edges will be oriented after an F turn, and the the number of oriented edges is changed by two. When exactly two edges on the F layer are oriented, an F turn changes the number of oriented edges by two, and if all or none edges on the F layer are oriented, the number of oriented edges does not change. Since all the edges of a solved cube are oriented and since slice turns and double-layer turns are just combinations of outer layer turns, all legal cube positions have an even number of oriented edges.

A corner is said to be an oriented corner or a corner of orientation 0, if it is located on the D layer and the sticker facing down has U or D color, or if it is located in the U layer and the sticker facing up has U or D color. A corner has orientation 1, if twisting it counterclockwise would leave it oriented correctly, and it has orientation 2, if twisting it clockwise would orient it.

Given two positive numbers a and n, a modulo n is the remainder of the division of a by n. For example, 5 modulo 3 is equal to 2. We write 5 = 2 mod 3. Now let a be the sum of the orientations of the eight corners. The remainder of the division of a by 3 is a number q between 0 and 2. We can write: a = q mod 3. If we now twist a single corner, the remainder of the division changes and we get a new remainder q'. It holds that q' = q + 1 mod 3 if the corner is twisted clockwise, and q' = q + 2 mod 3 if it is twisted counterclockwise.

Every outer layer turn twists two corners clockwise and two corners counterclockwise. So a single outer layer turn changes the remainder as follows: q' = q + 1 + 1 + 2 + 2 mod 3 = q + 6 mod 3 = q + 0 = q. So the remainder does not change at all. This means that all legal cube positions have the same remainder q. Since q = 0 if the cube is solved, we now know that the remainder of a legal cube position is always zero, i.e. the sum of the orientations of the eight corners is divisible by 3. This means that if you know the orientation of seven corner pieces of a legal cube position, the orientation of the eigth corner piece is uniquely determined.

A sequence of turns is called an algorithm. An example for an algorithm is the Sune algorithm: R U R' U R U2 R', which is the algorithm that starts with an R turn followed by a U turn and so on. After the Sune algorithm is applied to a cube, the D and E layers are unchanged, the orientation of the U layer edges is unchanged and the positions of the U layer corners relative to each other are unchanged. The algorithm only has an effect on the orientation of the U layer corners, on the position of the U layer edges and it has the same effect as a U2 turn on the position of the U layer corners. Let's focus on Sune's effect on the U layer corners. Sune does not change the orientation of the corner in ULF, twists the other three U layer corners clockwise and changes their positions as a U2 turn would.

The inverse of an algorithm A is the sequence of turns that undoes A. The inverse of the Sune algorithm, also called Anti-Sune, is: R U2 R' U' R U' R'. It changes the positions of the U layer corners like a U2 turn would and twists every U layer corner counterclockwise except the one that is in URB (because the corner in URB is in ULF after a U2 turn).

The mirror of the Sune algorithm across the S slice, also called Back-Sune, is: R' U' R U' R' U2 R. It twists every corner anticlockwise except ULB and changes their positions like a U2 turn would. The mirror of the Sune algorithm across the M slice is: L' U' L U' L' U2 L. The mirror of the Sune algorithm across the E slice is: R' D' R D' R' D2 R.

The inverse of an algorithm A is the sequence of turns that undoes A. The inverse of the Sune algorithm, also called Anti-Sune, is: R U2 R' U' R U' R'. It changes the positions of the U layer corners like a U2 turn would and twists every U layer corner counterclockwise except the one that is in URB (because the corner in URB is in ULF after a U2 turn).

The mirror of the Sune algorithm across the S slice, also called Back-Sune, is: R' U' R U' R' U2 R. It twists every corner anticlockwise except ULB and changes their positions like a U2 turn would. The mirror of the Sune algorithm across the M slice is: L' U' L U' L' U2 L. The mirror of the Sune algorithm across the E slice is: R' D' R D' R' D2 R.

Place your thumbs on the front face. Place your left thumb on the LF edge and your right thumb on the RF edge positions. Place your middle fingers on the LB and RB edge positions and your ring fingers on the DBL and DBR corner positions.

U - pull with right index finger or push with left index finger.

U' - pull with left index finger or push with right index finger.

U2 - right index finger followed by right middle finger. Or left index finger followed by left middle finger.

R, R' - rotate your wrist ('wrist turning'). Or do whatever you like, the R layer is very easy to turn.

R2 - wrist turning. Or right ring finger followed by right middle finger.

L, L', L2 - similar to R layer.

D - left ring finger.

D' - right ring finger.

F - right index finger or left thumb.

F' - right thumb or left index finger.

F2 - regrip the cube and use right index followed by right middle finger (or left index followed by left middle finger).

B, B', B2 - regrip the cube such that the B face faces upwards, then turn like you would turn the U layer.

M - right ring finger.

M' - right ring finger or right thumb or left index finger.

M2 - right ring finger followed by right middle finger.

E turns - Place your left thumb on the ULF corner, your left middle finger on the ULB corner, your right thumb on the DRF corner and your right ring finger on the DRB corner.

S turns - regrip the cube such that the F face faces upwards, then turn like you would turn the E layer.

U - pull with right index finger or push with left index finger.

U' - pull with left index finger or push with right index finger.

U2 - right index finger followed by right middle finger. Or left index finger followed by left middle finger.

R, R' - rotate your wrist ('wrist turning'). Or do whatever you like, the R layer is very easy to turn.

R2 - wrist turning. Or right ring finger followed by right middle finger.

L, L', L2 - similar to R layer.

D - left ring finger.

D' - right ring finger.

F - right index finger or left thumb.

F' - right thumb or left index finger.

F2 - regrip the cube and use right index followed by right middle finger (or left index followed by left middle finger).

B, B', B2 - regrip the cube such that the B face faces upwards, then turn like you would turn the U layer.

M - right ring finger.

M' - right ring finger or right thumb or left index finger.

M2 - right ring finger followed by right middle finger.

E turns - Place your left thumb on the ULF corner, your left middle finger on the ULB corner, your right thumb on the DRF corner and your right ring finger on the DRB corner.

E' - right index finger.

E - left ring finger.

E2 - right index followed by right middle finger.

M - pull with left index finger.

M' - push UF edge piece with left index finger.

E - left ring finger.

E2 - right index followed by right middle finger.

M - pull with left index finger.

M' - push UF edge piece with left index finger.

S turns - regrip the cube such that the F face faces upwards, then turn like you would turn the E layer.

move count - the number of outer layer turns needed to solve the cube or to solve a substep of a solving method. Turns like R, R', R2, r, r', r2 count as one move, but M, M', M2 count as two moves. If you want slice turns to count as one move as well, you have to change the metric from the htm metric (half turn metric) to the stm metric (slice turn metric).

A block is a solved part of the cube that can be located anywhere on the cube and that has dimension axbxc, for a, b and c in the set {1, 2, 3}. The whole cube is a 3x3x3 block when it is solved. A 1x1x2 block (or a 1x2x1 or 2x1x1 block) is called a pair. Pairs contain one edge piece and either a corner piece or a center piece. A line is a solved part of the cube that is a 1x1x3 block (or a 1x3x1 or 3x1x1 block).

home position - The home position of a piece or block is the position the piece or block will be located at when the cube is solved.

solving a piece or block - bring the cube in a cube position where the piece or block is located in its home position.

EO - Orientation of the edges, i.e. making sure there are no bad edges anywhere on the cube.

CP - Permutation of the corners, i.e. changing the positions of the corner pieces relative to each other.

CO - Orientation of the corners.

EP - Permutation of the edges.

F2L - Bring the cube in a cube position where the D and E layers are solved.

L6E - Starting from a cube position where the cube is solved except for the 6 edges located on the U and M layers, bring the cube into the solved state, i.e. solve the last six edges.

base pair - The pair consisting of the DL edge piece and the DBL corner piece.

base corner - The DFL corner piece.

base line - The line consisting of the base pair and the base corner.

A block is a solved part of the cube that can be located anywhere on the cube and that has dimension axbxc, for a, b and c in the set {1, 2, 3}. The whole cube is a 3x3x3 block when it is solved. A 1x1x2 block (or a 1x2x1 or 2x1x1 block) is called a pair. Pairs contain one edge piece and either a corner piece or a center piece. A line is a solved part of the cube that is a 1x1x3 block (or a 1x3x1 or 3x1x1 block).

home position - The home position of a piece or block is the position the piece or block will be located at when the cube is solved.

solving a piece or block - bring the cube in a cube position where the piece or block is located in its home position.

EO - Orientation of the edges, i.e. making sure there are no bad edges anywhere on the cube.

CP - Permutation of the corners, i.e. changing the positions of the corner pieces relative to each other.

CO - Orientation of the corners.

EP - Permutation of the edges.

F2L - Bring the cube in a cube position where the D and E layers are solved.

L6E - Starting from a cube position where the cube is solved except for the 6 edges located on the U and M layers, bring the cube into the solved state, i.e. solve the last six edges.

base pair - The pair consisting of the DL edge piece and the DBL corner piece.

base corner - The DFL corner piece.

base line - The line consisting of the base pair and the base corner.

Step 0 - Don't rotate the cube during the whole solve.

Step 1 (EOPair) - Solve the EO and bring the base pair to its home position.

Step 2 (CPLine) - Using only R, U and F2 turns, bring the cube into a cube position whose corners can be solved using only R and U turns, and simultaneously place the base corner in its home position, solving the base line.

Step 3 (Pseudo-block) - Using only R, U, r2 and u2 turns, solve the line consisting of the LF edge, the LB edge, and the left center, and either place the UF and UB edges in the DF and DB positions or place them in the DB and DF positions or place the UL and UR edges in the DF and DB positions or place them in the DB and DF positions.

Step 4 (COLine)

Step 4a - Using only R and U turns, place any two corners oriented on DR and place any one of the E slice edge pieces in the RF or RB position.

Step 4b - Use one of 34 algorithms or their mirror across the S slice, that orients all the corners and that places the last E slice edge piece in the E slice (not necessarily in its home position).

Step 5 (Pseudo-F2L) - Using only R2 and U turns, solve the line consisting of the DR corners and DR edge in such a way, that the line consisting of the RF edge, RB edge and right center is solved too.

Step 6 (L6E)

Using only M and U turns, solve the six edges of the U and M layers and the centers of the M layer.

Step 1 (EOPair) - Solve the EO and bring the base pair to its home position.

Step 2 (CPLine) - Using only R, U and F2 turns, bring the cube into a cube position whose corners can be solved using only R and U turns, and simultaneously place the base corner in its home position, solving the base line.

Step 3 (Pseudo-block) - Using only R, U, r2 and u2 turns, solve the line consisting of the LF edge, the LB edge, and the left center, and either place the UF and UB edges in the DF and DB positions or place them in the DB and DF positions or place the UL and UR edges in the DF and DB positions or place them in the DB and DF positions.

Step 4 (COLine)

Step 4a - Using only R and U turns, place any two corners oriented on DR and place any one of the E slice edge pieces in the RF or RB position.

Step 4b - Use one of 34 algorithms or their mirror across the S slice, that orients all the corners and that places the last E slice edge piece in the E slice (not necessarily in its home position).

Step 5 (Pseudo-F2L) - Using only R2 and U turns, solve the line consisting of the DR corners and DR edge in such a way, that the line consisting of the RF edge, RB edge and right center is solved too.

Step 6 (L6E)

Using only M and U turns, solve the six edges of the U and M layers and the centers of the M layer.

1. In the Line Method, pieces that are viewed as solved pieces are never removed from their home positions again, not even during the execution of an algorithm.

2. Neatness - After steps 1 and 2, the cube position is in the <R, U, r2, u2> group. This greatly reduces the complexity of the rest of the solve, and the way the Line Method is structured leads to a very pleasant solve from step 3 on. It would be tempting to quickly bring the cube into the <R, U> group by completely solving the 2x2x3 block that includes the base line, instead of just building a pseudo-block like in step 3. But I believe that the best way of solving the U layer edge pieces is by using M turns like in step 6, so there is no need to bring the cube into the <R, U> group.

3. Ergonomy - The Line Method is structured in a way that makes it possible to solve the cube while only using turns that are easy to perform, like R and U turns. Here is an overview:

step | turns used

2 | R, U, F2

3 | R, U, r2, u2 (if you want M2, E2)

4 | R, U

5 | R2, U

6a | M2, U

6b | M, U2

Also, there are no cube rotations in the Line Method.

4. Being suitable for speedsolving - Because of principle 1, the average move count needed to solve the cube with the Line Method is very low, and principle 3 makes it easy to achieve a high TPS (turns per second) without needing to learn any 'finger tricks' to speed up awkward move sequences.

5. Ease of learning - To be able to perform step 4b, it is required to learn 34 algorithms with an average move count of 8.2 and to learn in which cases each algorithm should be applied. In step 2 you need to be able to distinguish 6 cases and learn 3 algorithms with a move count of 5.

The downside of the Line Method w.r.t. speedsolving is that in step 2, you have to detect the positions of certain corners. This generally takes a short moment. This is reason enough to believe that the established speedsolving methods like CFOP, Roux or ZZ are probably better methods if execution speed is all you care for.

2. Neatness - After steps 1 and 2, the cube position is in the <R, U, r2, u2> group. This greatly reduces the complexity of the rest of the solve, and the way the Line Method is structured leads to a very pleasant solve from step 3 on. It would be tempting to quickly bring the cube into the <R, U> group by completely solving the 2x2x3 block that includes the base line, instead of just building a pseudo-block like in step 3. But I believe that the best way of solving the U layer edge pieces is by using M turns like in step 6, so there is no need to bring the cube into the <R, U> group.

3. Ergonomy - The Line Method is structured in a way that makes it possible to solve the cube while only using turns that are easy to perform, like R and U turns. Here is an overview:

step | turns used

2 | R, U, F2

3 | R, U, r2, u2 (if you want M2, E2)

4 | R, U

5 | R2, U

6a | M2, U

6b | M, U2

Also, there are no cube rotations in the Line Method.

4. Being suitable for speedsolving - Because of principle 1, the average move count needed to solve the cube with the Line Method is very low, and principle 3 makes it easy to achieve a high TPS (turns per second) without needing to learn any 'finger tricks' to speed up awkward move sequences.

5. Ease of learning - To be able to perform step 4b, it is required to learn 34 algorithms with an average move count of 8.2 and to learn in which cases each algorithm should be applied. In step 2 you need to be able to distinguish 6 cases and learn 3 algorithms with a move count of 5.

The downside of the Line Method w.r.t. speedsolving is that in step 2, you have to detect the positions of certain corners. This generally takes a short moment. This is reason enough to believe that the established speedsolving methods like CFOP, Roux or ZZ are probably better methods if execution speed is all you care for.

Beginners can solve EO first (using any face turns), then, using only L, R, U, D, F2 and B2 turns, pair up the DL edge and DBL corner somewhere on the cube and place this pair in its home position. 180 degree slice turns and 180 degree double-layer turns also preserve EO and are thus allowed.

To solve EO, locate the bad edges as described in chapter 2. An F, F', B or B' turn causes the good edges on the F resp. B layer to become bad edges and vice versa. Strategy for solving EO:

2 bad edges: Using only L, R, U and D turns, place exactly one bad edge on the F layer, then make an F or F' turn. You have 4 bad edges now (3 on the F layer). You can also do this on the B layer.

4 bad edges: Using only L, R, U and D turns, place all the bad edges on the F layer (or all on the B layer), then make an F or F' turn.

6 bad edges: Place exactly 3 bad edges on the F layer and make an F or F' turn. Or treat as 4 + 2.

8 bad edges: Treat as 4 + 4.

10 bad edges: Treat as 4 + 4 + 2. Keep track of the good edges rather than the bad edges.

12 Treat as 4 + 4 + 4.

To solve EO, locate the bad edges as described in chapter 2. An F, F', B or B' turn causes the good edges on the F resp. B layer to become bad edges and vice versa. Strategy for solving EO:

2 bad edges: Using only L, R, U and D turns, place exactly one bad edge on the F layer, then make an F or F' turn. You have 4 bad edges now (3 on the F layer). You can also do this on the B layer.

4 bad edges: Using only L, R, U and D turns, place all the bad edges on the F layer (or all on the B layer), then make an F or F' turn.

6 bad edges: Place exactly 3 bad edges on the F layer and make an F or F' turn. Or treat as 4 + 2.

8 bad edges: Treat as 4 + 4.

10 bad edges: Treat as 4 + 4 + 2. Keep track of the good edges rather than the bad edges.

12 Treat as 4 + 4 + 4.

Using only R and U turns, place the base corner in the UBL position, and place the two DR corners in the DR positions. The exact position of the DR corners is not important, but you need to remember if the DR corners are in their home positions or if they are swapped. Corner orientation is completely irrelevant for the placement of the three corners.

Number the U layer corner pieces clockwise, e.g. UBL = 1, UBR = 2, UFR = 3, UFL = 4. Now assess the current state of the corners in the U layer positions. We know that the base corner is located in the UBL position and that the other corner pieces currently in the U layer are three U layer corner pieces, to which we just assigned numbers. As an example, there could be the corners 3, 1 and 4 in the UBR, UFR and UFL positions, in this order (the order is important here). So we have an ordered triplet of numbers (a, b, c) where a is the corner piece in position UBR, b the corner piece in position UFR, and c the corner piece in position UFL. a, b, and c are numbers between 1 and 4.

We say that there is a forward relation between a and b, if (a,b) is an element of the set { (1,2), (2,3), (3,4), (4,1) }. There is a backward relation between a and b, if (a,b) is in { (4,3), (3,2), (2,1), (1,4) }, and there is a jump relation between a and b, if (a,b) is in the set { (1,3), (3,1), (2,4), (4,2) }. Similarly the corner pieces b and c can be in a forward (fw), backward (bw) or jump (j) relation. Now we transform the ordered triplet of numbers into an ordered pair of relations, e.g. (a,b,c) = (3,1,4) becomes (jump, backward), or (j, bw). Don't forget that a ist the corner piece in position UBR et cetera.

Now we have to remember if the DR corner pieces are currently in their home positions or if they are swapped. In case they are swapped, transform your ordered pair of relations as follows: forward becomes backward, backward becomes forward, and a jump relation stays a jump relation. So if the DR corner pieces are swapped, (3,1,4) has (j, fw) relations.

In case we end up with the (fw, fw) pair, we apply one of three possible algorithms to the cube and after that we are done with step 2 and go on to step 3.

If the base corner is oriented (i.e. if the D color is facing upwards), we apply the algorithm F2 U' R U' F2.

If the base corner has orientation 1 (i.e. if the F color is facing upwards), we apply the algorithm F2 U' F2 R F2.

If the base corner has orientation 2 (i.e. if the L color is facing upwards), we apply the algorithm F2 U R' U' R' F2.

In case we don't have the (fw, fw) pair, we must find the "swap corner". The swap corner is the corner piece that is currently located in the following position:

relations | swap corner location | swap corner

(fw, j) | UFL | last corner piece in the triplet

(j, fw) | UBR | first corner piece in the triplet

(bw, bw) | UFR | middle corner piece in the triplet

(bw, j) | DBR | either DBR piece or DFR piece

(j, bw) | DFR | either DBR piece or DFR piece

Now, using only R and U turns, place the swap corner in the UBR position and place the base corner oriented in the UFR position, then do an F2 turn to complete the base line. The F2 turn performs the "corner swap" that brings the cube in a cube position whose corners can be solved using only R and U turns.

Number the U layer corner pieces clockwise, e.g. UBL = 1, UBR = 2, UFR = 3, UFL = 4. Now assess the current state of the corners in the U layer positions. We know that the base corner is located in the UBL position and that the other corner pieces currently in the U layer are three U layer corner pieces, to which we just assigned numbers. As an example, there could be the corners 3, 1 and 4 in the UBR, UFR and UFL positions, in this order (the order is important here). So we have an ordered triplet of numbers (a, b, c) where a is the corner piece in position UBR, b the corner piece in position UFR, and c the corner piece in position UFL. a, b, and c are numbers between 1 and 4.

We say that there is a forward relation between a and b, if (a,b) is an element of the set { (1,2), (2,3), (3,4), (4,1) }. There is a backward relation between a and b, if (a,b) is in { (4,3), (3,2), (2,1), (1,4) }, and there is a jump relation between a and b, if (a,b) is in the set { (1,3), (3,1), (2,4), (4,2) }. Similarly the corner pieces b and c can be in a forward (fw), backward (bw) or jump (j) relation. Now we transform the ordered triplet of numbers into an ordered pair of relations, e.g. (a,b,c) = (3,1,4) becomes (jump, backward), or (j, bw). Don't forget that a ist the corner piece in position UBR et cetera.

Now we have to remember if the DR corner pieces are currently in their home positions or if they are swapped. In case they are swapped, transform your ordered pair of relations as follows: forward becomes backward, backward becomes forward, and a jump relation stays a jump relation. So if the DR corner pieces are swapped, (3,1,4) has (j, fw) relations.

In case we end up with the (fw, fw) pair, we apply one of three possible algorithms to the cube and after that we are done with step 2 and go on to step 3.

If the base corner is oriented (i.e. if the D color is facing upwards), we apply the algorithm F2 U' R U' F2.

If the base corner has orientation 1 (i.e. if the F color is facing upwards), we apply the algorithm F2 U' F2 R F2.

If the base corner has orientation 2 (i.e. if the L color is facing upwards), we apply the algorithm F2 U R' U' R' F2.

In case we don't have the (fw, fw) pair, we must find the "swap corner". The swap corner is the corner piece that is currently located in the following position:

relations | swap corner location | swap corner

(fw, j) | UFL | last corner piece in the triplet

(j, fw) | UBR | first corner piece in the triplet

(bw, bw) | UFR | middle corner piece in the triplet

(bw, j) | DBR | either DBR piece or DFR piece

(j, bw) | DFR | either DBR piece or DFR piece

Now, using only R and U turns, place the swap corner in the UBR position and place the base corner oriented in the UFR position, then do an F2 turn to complete the base line. The F2 turn performs the "corner swap" that brings the cube in a cube position whose corners can be solved using only R and U turns.

This step can be done using only R, U, r2 and u2 turns, but feel free to also use M2, E2 turns or sequences like M' U2 M, that preserve EO. Build the line consisting of the LF and LB edges and the left center on the R layer of the cube and insert it to its home position with a u2 or E2 turn. Place the UF and UB edges (or UL and UR edges) opposite of each other on the U layer and insert them on the D layer with an r2 or M2 turn. Alternatively, place the edges diagonally across from each other on the M slice and do M' U2 M or M U2 M'. You can solve the line before you place the two U edges or after, it doesn't matter.

This step is done using only R and U turns. Step 4a: Unless there is already an E slice edge in the E slice with an oriented corner underneath the edge, build a "pair" on the U layer consisting of one of the E layer edges and any of the six unsolved corners. The U or D sticker of the corner pieces should be facing away from the E layer edge. Depending on if the corner is on the left or on the right of the edge, insert the "pair" in the FR/DFR position with a single R' turn or in the BR/DBR position with an R turn. Then insert any corner in the other DR corner position in such a way that it is oriented, unless the corner in the DR position is oriented already. There are many other ways to solve step 4a, in general it is a good idea to try to insert the "pair" and corner simultaneously.

Step 4b: The following list of 34 algorithms solves step 4b if an E layer edge was paced in the BR position in step 4a. In the algorithm list, "edge" stands for the E layer edge piece we want to insert into the E layer, and the 2x2 matrix stands for the current orientations of the corners located in the U layer:

2 1

0 0

This matrix is supposed to mean that the corner in the UBL position has orientation 2 (i.e. twisting it clockwise would orient it), the corner in UBR position has orientation 1, and the other corners currently in the U layer are oriented correctly. You have to detect where your E layer edge, that you want to insert into the FR position, is located and how the corners are currently oriented, then apply the algorithm and go on to step 5.

If there is no E layer edge in the BR position, you have to mirror one of the algorithms across the S slice and apply this mirrored algorithm to the cube. Here are some examples that show how to do this:

Example 1: For the case: Pi, edge in UR, you pick the algorithm that is listed under Pi, edge in UL, but you prepend a U2 turn at the start of the algorithm and you replace every R turn of the algorithm with an R' turn and vice versa, and replace every U turn with a U' turn and vice versa.

Example 2: For the case: S, edge in UF, you pick the algorithm that is listed under AS, edge in UB, and you replace R with R' and vice versa, and U with U' and vice versa.

The 34 algorithms were chosen such that both the algorithm itself and the mirrored algorithm are easy to perform.

Beginners should place both E layer edges in the E layer during step 4a (using the strategy of pairing an E layer edge up with any corner as described in step 4a, inserting the pair, then pairing the other E layer edge up with a corner and inserting the pair), so they only have to learn 7 algorithms for the cases where all the E layer edges are in the E layer ("edge in place" cases).

O (Oriented)

0 0

0 0

edge in UL

R U' R' U2 R U R'

S (Sune)

2 2

0 2

edge in place

R U R' U R U2 R' (= Sune)

edge in UF

U R2 U2 R U R2 U2 R' U R

edge in UR

R2 U R2 U R U' R'

edge in UB

U R2 U2 R U' R' U R

edge in UL

R U' R' U R U2 R'

AS (Anti-Sune)

1 0

1 1

edge in place

U R U2 R' U' R U' R' (= Anti-Sune)

edge in UB

R2 U R2 U' R' U' R U R

edge in UL

R2 U R' U R'

edge in UF

U' R2 U2 R' U' R' U2 R U2 R'

edge in UR

R2 U2 R U' R2 U2 R U' R

H (CO looks like the letter H)

2 1

1 2

edge in place

U R' U' R2 U' R U2 R2 U' R'

edge in UF

U2 R2 U R' U' R2 U' R U R

edge in UR

R' U R' U2 R U' R' U2 R

Pi (CO looks like the greek letter Pi)

1 2

1 2

edge in place

U R U2 R2 U' R2 U' R2 U2 R (= Bruno)

edge in UF

R U2 R' U2 R U' R'

edge in UR

R2 U R' U2 R' U R U' R'

edge in UB

U' R' U' R U' R2 U2 R

edge in UL

R U R' U' R U2 R'

T (CO looks like the letter T)

1 2

0 0

edge in place

R' U R' U' R U R2 U2 R' U2 R'

edge in UF

R' U' R U' R2 U R' U R

edge in UR

R2 U R' U R2 U' R' U R

edge in UB

U R U2 R' U2 R U2 R'

edge in UL

R' U2 R' U R2 U' R U2 R

U (CO looks like an upside down U)

0 0

1 2

edge in place

U2 R U R' U' R U R' U' R U R' (= Triple Sexy)

edge in UB

R' U' R U R' U2 R' U' R'

edge in UL

U R U' R U R' U R'

edge in UF

U' R' U2 R' U R' U' R U2 R

edge in UR

R' U2 R' U2 R

L (CO looks like an upside down L)

1 0

0 2

edge in place

U2 R U R' U' R U2 R' U2 R U2 R'

edge in UB

U' R2 U2 R2 U' R' U2 R U2 R

edge in UL

U' R' U2 R2 U R U' R U2 R

edge in UF

R2 U' R' U2 R2 U2 R U2 R

edge in UR

R2 U' R U2 R' U2 R

Step 4b: The following list of 34 algorithms solves step 4b if an E layer edge was paced in the BR position in step 4a. In the algorithm list, "edge" stands for the E layer edge piece we want to insert into the E layer, and the 2x2 matrix stands for the current orientations of the corners located in the U layer:

2 1

0 0

This matrix is supposed to mean that the corner in the UBL position has orientation 2 (i.e. twisting it clockwise would orient it), the corner in UBR position has orientation 1, and the other corners currently in the U layer are oriented correctly. You have to detect where your E layer edge, that you want to insert into the FR position, is located and how the corners are currently oriented, then apply the algorithm and go on to step 5.

If there is no E layer edge in the BR position, you have to mirror one of the algorithms across the S slice and apply this mirrored algorithm to the cube. Here are some examples that show how to do this:

Example 1: For the case: Pi, edge in UR, you pick the algorithm that is listed under Pi, edge in UL, but you prepend a U2 turn at the start of the algorithm and you replace every R turn of the algorithm with an R' turn and vice versa, and replace every U turn with a U' turn and vice versa.

Example 2: For the case: S, edge in UF, you pick the algorithm that is listed under AS, edge in UB, and you replace R with R' and vice versa, and U with U' and vice versa.

The 34 algorithms were chosen such that both the algorithm itself and the mirrored algorithm are easy to perform.

Beginners should place both E layer edges in the E layer during step 4a (using the strategy of pairing an E layer edge up with any corner as described in step 4a, inserting the pair, then pairing the other E layer edge up with a corner and inserting the pair), so they only have to learn 7 algorithms for the cases where all the E layer edges are in the E layer ("edge in place" cases).

O (Oriented)

0 0

0 0

edge in UL

R U' R' U2 R U R'

S (Sune)

2 2

0 2

edge in place

R U R' U R U2 R' (= Sune)

edge in UF

U R2 U2 R U R2 U2 R' U R

edge in UR

R2 U R2 U R U' R'

edge in UB

U R2 U2 R U' R' U R

edge in UL

R U' R' U R U2 R'

AS (Anti-Sune)

1 0

1 1

edge in place

U R U2 R' U' R U' R' (= Anti-Sune)

edge in UB

R2 U R2 U' R' U' R U R

edge in UL

R2 U R' U R'

edge in UF

U' R2 U2 R' U' R' U2 R U2 R'

edge in UR

R2 U2 R U' R2 U2 R U' R

H (CO looks like the letter H)

2 1

1 2

edge in place

U R' U' R2 U' R U2 R2 U' R'

edge in UF

U2 R2 U R' U' R2 U' R U R

edge in UR

R' U R' U2 R U' R' U2 R

Pi (CO looks like the greek letter Pi)

1 2

1 2

edge in place

U R U2 R2 U' R2 U' R2 U2 R (= Bruno)

edge in UF

R U2 R' U2 R U' R'

edge in UR

R2 U R' U2 R' U R U' R'

edge in UB

U' R' U' R U' R2 U2 R

edge in UL

R U R' U' R U2 R'

T (CO looks like the letter T)

1 2

0 0

edge in place

R' U R' U' R U R2 U2 R' U2 R'

edge in UF

R' U' R U' R2 U R' U R

edge in UR

R2 U R' U R2 U' R' U R

edge in UB

U R U2 R' U2 R U2 R'

edge in UL

R' U2 R' U R2 U' R U2 R

U (CO looks like an upside down U)

0 0

1 2

edge in place

U2 R U R' U' R U R' U' R U R' (= Triple Sexy)

edge in UB

R' U' R U R' U2 R' U' R'

edge in UL

U R U' R U R' U R'

edge in UF

U' R' U2 R' U R' U' R U2 R

edge in UR

R' U2 R' U2 R

L (CO looks like an upside down L)

1 0

0 2

edge in place

U2 R U R' U' R U2 R' U2 R U2 R'

edge in UB

U' R2 U2 R2 U' R' U2 R U2 R

edge in UL

U' R' U2 R2 U R U' R U2 R

edge in UF

R2 U' R' U2 R2 U2 R U2 R

edge in UR

R2 U' R U2 R' U2 R

Using only R2 and U turns, build the DR line on the U layer starting with a pair consisting of the DR edge and one of the DR corners. To build this pair, use an R2 turn to place a DR corner in one of the DR positions, with the R color sticker either facing to the front or facing to the back, and place the DR corner in the U layer such that an R2 turn connects the corner and with the edge. Now do the same with the second DR corner piece to complete the line and align the line with the FR and BR edges s.t. an R2 turn solves the DR line as well as the line consisting of the FR and BR edges and the R center.

Step 6a - If you placed the UL and UR edges on the D layer in Step 3, solve the line consisting of the UL edge and UL corners as well as the line consisting of the UR edge and UR corners, using only M2 and U turns.

If you placed the UF and UB edges on the D layer in Step 3, build the line consisting of the UF edge and UF corners as well as the line consisting of the UB edge and UB corners using only M2 and U turns and place the two lines in the UL and UR positions.

Step 6b - Using only M and U2 turns, solve the remaining four edges and four centers (and adjust the U layer with a U or U' if you placed the UF and UB lines in the UL and UR positions in Step 6a).

If every edge piece is solved after step 6a, but the centers are not solved and the D center is in the U layer, it is better to use E2 turns to finish the solve, like this: E2 M' E2 M. Tips on how to perform E2 turns can be found in chapter 4.

If you placed the UF and UB edges on the D layer in Step 3, build the line consisting of the UF edge and UF corners as well as the line consisting of the UB edge and UB corners using only M2 and U turns and place the two lines in the UL and UR positions.

Step 6b - Using only M and U2 turns, solve the remaining four edges and four centers (and adjust the U layer with a U or U' if you placed the UF and UB lines in the UL and UR positions in Step 6a).

If every edge piece is solved after step 6a, but the centers are not solved and the D center is in the U layer, it is better to use E2 turns to finish the solve, like this: E2 M' E2 M. Tips on how to perform E2 turns can be found in chapter 4.