Background

The Strategy of Eight Corners was first developed by one of my classmates, Mr. Edward Leung. The basic rules of his original design include:

• edges are easier to move than corners
• edges on the middle slices (the three planes without corners) are the easiest to move
• the symmetry of the cube should be exploited

More rules have subsequently been discovered, and are used to reselect the algorithms used.

Nomenclature

This section deals with how to name things.

General

We define START as the position where each face of the cube is showing only one color. All new cubes arrive in the START position.

We usually start the cube in a random configuration. The aim is to solve the cube. That is, to restore it back to the configuration where each face shows only one color.

An algorithm is any sequence of movements which produce an interesting result (e.g. twisting a corner by 120°).

A strategy is a system of algorithms for solving the cube. It specifies the sequence of sub-goals to be attained, and the algorithm to be used in each situation. A strategy refers to many algorithms; an algorithm can be used in many strategies. The strategy presented here is the Strategy of Eight Corners.

Parts of the Cube

Names are assigned to parts of the cube. All names are relative to the spatial directions, irrespective of color of the faces.

From now on we assume that the cube is in front of us in such a way that the Front, Back, Right, Left, Up and Down faces can be clearly identified. The highlighted initial letters are the standard face names:

• Right (green),
• Left (blue),
• Up (yellow),
• Down (white),
• Front (red),
• Back (orange).

The colors specified above are the usual colors used in a physical cube on the market. It is also the color of the Roofpig cube displayed on these web pages.

Although they are perfectly symmetric, algorithms concentrate their movements on the R/D/U/F faces (in descending priority). Observations are nearly always made on F/U faces. Thus, the algorithms usually move faces that area easiest to move and observations are made on faces that are easiest to observe.

The twelve edges are called by the name of the two faces forming the edge: e.g. UR. When calling a single edge, UR can be called RU also.

The eight corners are called by the name of the three faces forming the corner: e.g. URF.

Movements

Rotations are named after the name of the face. A rotation X rotates the face X clockwise by 90°, viewed from the exterior of the cube towards the face X. Therefore, R moves RF to RU, and L moves LF to LD. R^-1 moves the R face in the opposite direction: it is the inverse of R.

We move the slices (the three central planes without corners) as well. Note that I am now using the turn names from Roofpig.

• The slice between R and L is called M, for Middle. M moves in the same direction as L: e.g., Middle moves UF to FD.
• The slice between U and D is called E, for Equatorial. E moves in the same direction as D: e.g., E moves LF to FR.
• The slice between F and B is called S, for Standing. S moves in the same direction as F: e.g., S moves LU to UR. S is rarely used because this slice is not easy to access.

Demonstrating the following moves: R, R^-1, M, M^-1, L, L^-1, U, U^-1, E, E^-1, D, D^-1, F, F^-1, S, S^-1, B, B^-1.

Notice that R and L actually rotate in opposite directions because of the way they are defined. Click the triangular arrow to see each move in turn.

I find that the direction of the slice movements, M and E, are really opposite of what I expected, but this is the Roofpig convention and I have to stick with it.

In the Strategy of Eight Corners, we focus a lot on the U face. We also turn the cube upside down from time to time, treating the other face as the top. That is why we chose the pair of faces with brightest colors (white and yellow) as U and D.

Sometimes, we rotate the whole cube to see it from another angle. I find that the X/Y/Z used in the standard notion too difficult to follow. As a result, I defined a new set of names for them:

• Ra ≡ X
• Ua ≡ Y
• Fa ≡ Z
• La ≡ X^-1
• Da ≡ Y^-1
• Ba ≡ Z^-1

Twists and Flips

A corner at a certain place can have three states. Algorithms that take a corner from one state to another will twist the corner. For pratical reasons, twisting is considered undefined when a corner does not stay at the same place. Mathematically, twisting can still be defined, but only in a less intuitive way and is of little concern to us.

When viewed from the exterior toward the corner, a clockwise twist of 120° is called positive (e.g. URF → RFU, or simply X → X₊). Twisting in the opposite direction is called negative (e.g. URF → FUR, or simply X → X_-). Note that X₊₊ ≡ X_-, and X₊₊₊ ≡ X.

An edge at a certain place can have two states. Algorithms that take an edge from one state to another will flip the edge. Flipping is considered undefined when an edge is moved to another place. An example of flipping is UR → RU, or simply X → X₊. Note that X₊₊ ≡ X.

Inverse

If X is an algorithm, then X^-1 applies X backwards, and making all face movements in the opposite direction.

If X = Y⊕Z (i.e. Y followed by Z), X^-1 = Z^-1⊕Y^-1

Color

The moves are themselves color-independent, but at times it is useful to refer to color names for easy reference to the cube displayed.

The original version of this guide used colors assignments that looks strange to me now. I am not sure whether it was to align with Neil's cube, or whether I really was using such weird colors. During the rewrite to use the Roofpig cube, I changed all color references to align with theRoofpig cube, which fortunately coincides with my Rubiks Cube.

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