Permutation Notation

The permutation notation is used to describe location and orientation changes of cube parts.

Components of a Rubik's Cube:

A Rubik's Cube has 6 faces.

It has 6 side parts, 8 corner parts and 12 edge parts.

Side parts have 1 sticker, edge parts have 2 stickers, corner parts have 3 stickers.

Describing the location of a part:

The location of a part can be described by listing the faces on which the stickers of the part can be seen.

e.g. The faces can be denoted by the letters: «r», «u», «f», «l», «d», «b» (right, up, front, left, down, back).

e.g. «r» describes that the sticker of a side part can be seen from the right of the cube.

e.g. «fu» describes that the stickers of an edge part can be seen at the front and upper faces of the cube.

e.g. «frd» describes that the stickers of a corner part can be seen at the front, right, and down faces of the cube.

Describing location changes of parts:

Location changes always happen in cycles. If a part moves into the location of another part, then that part must move as well. This may trigger more location changes until one part closes the cycle by moving into the start location of the cycle.

To describe a cycle, we list the location of the parts in the sequence of the change. Each cycle is enclosed in brackets. Each location description is separated by a comma.

e.g. «(fu,fr,fd,fl)» describes, that the edge part at «fu» moves to «fr», «fr» moves to «fd», «fd» moves to «fl», and «fl» moves to «(fu)».

e.g. «(fu,fd) (lf,rf)» denotes two disjoint cycles. «fu» and «fd» are swapping their locations, and «lf» and «rf» are swapping their locations as well.

Describing orientation changes of parts:

When the orientation of a part changes, then it stickers can be seen in a different sequence on the faces of the cube.

The face names in a cycle can be used to describe orientation changes as well. The first face name in a location description denotes where the first sticker goes. The second face name in a location description denotes where the second sticker goes, and so on.

e.g. «(fu,fr,...)» describes, that the first sticker on face «f» at edge location «fu» goes to face «f» at edge location «fr», and the second sticker on face «u» at location «fu» goes to face «r» at location «fr».

e.g. «(ufl,rfu,...)» describes, that the first sticker on face «u» at corner location «ufl» goes to face «r» at corner location «rfu», the second sticker on face «f» at corner location «ufl» goes to face «f» at «rfu», and the third sticker on «l» at «ufl» goes to «u» at «rfu».

Describing the orientation change of the last part in a cycle

To denote an orientation change of the last edge part in a cycle, we prepend a «+» to the location description of the first part of the cycle.

To denote an orientation change in clockwise direction of the last corner part in a cycle, we prepend a «+» to the location description of the first part in the cycle. To denote an orientation change in anticlockwise direction, we prepend a «-».

e.g. «(+fu,fr)» describes that the sticker on face «f» of the last part «fr» goes to face «u» of the first part «fu», and the sticker «r» of «fr» goes to «f» at «fu».

Describing orientation changes of side parts:

A side part can have four different orientations. Since we can not tell the orientation of a side part from its location name, we have to use a prefix. No prefix denotes that the side part is in its default orientation. «+» denotes that the side part is oriented at 90° in clockwise direction. «++» denotes that the side part is oriented at 180°. «-» denotes that the side part is oriented at 90° in anticlockwise direction.

An orientation change is denoted by the difference between the orientation of the part at the current location and the orientation of the part at the next location.

To describe the orientation change of the last part in a cycle, we prepend an orientation token to the first part in the cycle.

e.g. «(f,+r)» describes that the side part at location «f» moves to «r» and changes its orientation by 90° in clockwise direction. The side part at «r» moves to «f» and changes its orientation by 90° in anticlockwise direction.

e.g. «(+f,+r)» describes that the side part at location «f» moves to «r» and changes its orientation by 90° in clockwise direction. The side part at «r» moves to «f» without changing its orientation.