EOLine

This stage orients all edges on the cube (EO) while simultaneously placing the DF and DB edges (Line). This reduces the cube to the <L,R,U> group, meaning the rest of the cube can be solved by turning only the L, R and U faces.

EOLine takes an average of ~6.127 moves and a maximum of 9 moves,[2] but is without doubt the most difficult part of the ZZ method. For this reason, it is often divided into two sub-stages: (1) Edge orientation followed by (2) placement of the line edges. Also known as "EO+Line".

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Edge Orientation Detection

An edge is defined as oriented if it can be solved using only R, L, U and D face turns. If an edge cannot be solved using these face turns then it is a misoriented or 'bad' edge.


All visible edges on this cube are oriented.

All visible edges on this cube are misoriented.



The following will tell you if you have a misoriented 'bad' edge. Where colours are stated it is assumed that the cube is held with white/yellow on top and green/blue on front.

  1. Look at the U/D faces. If you see:

    • L/R colour (orange/red) it's bad.
    • F/B colour (green/blue) means you need to look round the side of the edge. If the side is U/D (white/yellow) it is bad.


  2. Then look at the F/B faces of the E-slice (middle layer). The same rules apply. If you see:

    • L/R colour (orange/red) it's bad.
    • F/B colour (green/blue) means you need to look round the side of the edge. If the side is U/D (white/yellow) it is bad.

Using this method, look at each edge in the U-layer one-by-one, positively identifying any bad edges. Then look at the D-layer, applying the same rules to each edge. Finally look at the four remaining E-slice edges, remembering to scan the F/B faces of these edges first, applying the same rule. Keep track of bad edges by placing a finger on each one found.

Edge Orientation Strategy

A quarter turn of the F or B face causes its oriented edges to become misoriented, and its misoriented edges to become oriented. Using F/B quarter turns to orient up to four edges at a time is the core technique behind edge orientation. NOTE: F2 and B2 turns do not affect edge orientation. The following is a basic strategy for orienting edges, based on the number of misoriented edges on the cube.

These rules of thumb should help guide your decision making when manipulating bad edges:

  1. It is generally more efficient to chose the F/B face with most bad edges already on it for your initial F/B turn.
  2. It is generally less efficient if placement of a bad edge into F/B removes another bad edge from F/B.
  3. In cases with six or more bad edges, it is generally more efficient to eliminate the bad edges using both F and B faces rather than just one.
  4. In cases with six or more bad edges, it is generally more efficient if placement of a bad edge into a F/B face also places another edge into the opposite F/B face.
  5. Execution of F/B moves with only two opposite bad edges placed is futile from an edge orientation perspective, but can be useful for manipulation of line edges.

Edge Orientation Cases

Because there are over 2000 distinct edge orientation cases, it would be impractical to memorise an algorithm for each one. Instead, learn to recognise familiar EO patterns and the techniques which can be used to solve them.

In the vast majority of solves, edge orientation is carried out in groups of four. For this reason, a range of four-edge cases is presented below, along with a complete set of techniques to place them into the F-face. These techniques can just as easily be reflected to apply to the B-face. Many cases have more than one optimal solution, allowing a degree of freedom to help with placing edges on the opposite face, or setting up the line edges.

Placing edges individually

Using F2/B2 to create openings

Moving three edges simultaneously

Moving two edges simultaneously

Combining 2-edge move with F2

Removing placed edges

Combining edge removal and F2

Solving the Line

Once all the edges have been oriented, search for the DF and DB edges and solve them intuitively, being careful not to use F or B moves. Solving the line normally takes 1 to 3 moves and may require 4 moves in some rare cases.

Eventually, after a bit of practice on edge orientation it is possible to start thinking about integrating the line into EO to solve it more efficiently and without looking. More details on this appear in the full version of the tutorial.

Example EOLine Solves

  1. Scramble: F U2 D2 B2 U' F U' R' L' D R'

    Observations: This case has 4 bad edges in positions UF, UL, DL and DB. The basic edge orientation strategy will place all 4 edges into an F/B side and do an F/B turn. The DF line edge is in the DR position and is oriented. The DB edge is in the LU position and is misoriented.

    EOLine: D' L' R' U2 B

    Explanation: The initial D' move placed the oriented DF edge into its line position. The L' R' U2 moves placed all misoriented edges into the B-face, and placed the misoriented DB edge into a group 1 position. The final B turn oriented the 4 bad edges and completed the line.

  2. Scramble: F' B' U L2 D B R2 L2 U'

    Observations: 8 misoriented edges, elimination in two groups of 4. Four bad edges already present in F face. DB and DF both in group 1 positions in B face

    EOLine: F D B U2 F2

    Explanation: The initial F turn immediately removes 4 bad edges to leave 4 remaining. The D turn positions the only remaining bad edge not already in the B face. The B turn orients the last 4 edges, leaving the DF and DB edges in group 1 positions so that the line can be completed with U2 F2.

  3. Scramble: R F2 B' D' L U2 L2 D2 B

    Observations: 6 misoriented edges. Most likely elimination as 3 + 3 since generally more efficient. Both DF and DB edges are misoriented. DF edge in group 1 position, while DB edge is close to a group 1 position.

    EOLine: B' U2 D' L' B F2

    Explanation: The initial B' turn removes 3 bad edges to leave 4 remaining for elimination. The following U2 turn does two things: Places a bad edge and moves DF closer to its group-1 goal position. D' and L' then place 2 more bad edges into the B-face, while placing the DB edge into a group-1 position. B orients the last 4 bad edges and places the DB edge, and finally F2 places the DF edge to complete the line.

  4. Scramble: R' D F2 R B' U F' D' F2

    Observations: 8 edges are misoriented, so the basic orientation strategy will be to eliminate 4 edges, twice. The F-face has 3 bad edges, which is more than the B-face which has 2. The DB edge is in its correct position and oriented - also a group 1 position. The DF edge is in its correct position but misoriented - a group 2 position.

    EOLine: U F L B' D U B

    Explanation: The initial U turn fills the F-face with bad edges and F eliminates them. At this point the DF-edge is in a group 2 position, while the DB-edge is in a group 1 position. The following L-turn brings the DF-edge into a group 1 position, but now both line edges now perpendicular to each other in the D-face. For this reason a B' turn is made to move the DB edge out of the way and allow the DF-edge to be placed. Meanwhile the number and location of bad edges is unaffected. The following D and U turns place the 2 remaining bad edges, as well as the DF-edge. The final B-turn orients the last 4 bad edges and places the DB-edge.


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Last updated: 7th August 2016