# Mag's Rubik's Cube Notation

## Why you are here

This document describes the notation I use in describing operations on Rubik's Cube type puzzles. I use this notation in my 3x3x3 cube solution, my 4x4x4 cube solution, and my 5x5x5 cube solution.

My notation bears similarities to and differences from several other somewhat well-known notations. It is very simple, yet it has advantages over some other notations:

1. It covers cubes of any size. (Actually, for 10x10x10 and beyond, it would need a modification. But these cubes don't/can't exist in real life anyway.) It is easy to describe turns of inner slices on higher-order cubes, as well as turns of multiple slices at the same time.
2. It covers re-orientations of the entire cube, such as "roll the cube toward you, so that the top face comes the front".
3. It is succinct -- even in the 3x3x3 case, most operations can be described with equal or fewer characters than in most other notations; some operations can be described with much fewer.

## What you came for

A move consists of a face, a slice, and a direction. Each part is represented by a single character.

Faces are the 6 sides of the cube. These are named Up, Down, Left, Right, Front, and Back, and are represented by their initial letters, U, D, L, R, F, and B

Slices, sometimes called slabs, are the divisions of a cube due to planar cuts. On the 3x3x3 cube, there are three horizontal slices parallel to the U and D faces, three vertical slices parallel to the L and R faces, and three vertical slices parallel to the F and B faces.

The slices are numbered 1..N relative to a face. For example, the horizontal slices on a 4x4x4 cube, from top to bottom, are U1, U2, U3, and U4. The same four slices are also called D4, D3, D2, and D1, respectively, because they are the 4th, 3rd, 2nd, and 1st from the bottom. (This is just one of many examples of redundancy in my notation. Oh well.)

The direction of a move can be +, which means a quarter turn clockwise, -, which means a quarter turn counterclockwise, or *, which means a half turn.

# Glossary

Cubelet (also Cubie)
An individual colored piece of the cube. There are 26 cubelets on the 3x3x3 cube, 56 on the 4x4x4, and 98 on the 5x5x5. On the 3x3x3 and 5x5x5, six cubelets remain fixed (except for rotation) on the faces of the cubes. The rest of the cubelets move about the cube freely, to our great enjoyment and consternation.
• 3x3x3 cubelets
• 8 corners
• 12 edges
• 6 middles
• 4x4x4 cubelets
• 8 corners
• 24 edge pieces
• 2 of these make an "edge"
• 24 middle pieces
• 4 of these make an "middle")
• 5x5x5 cubelets
• 8 corners
• 36 edge pieces: 12 edge-middles, 24 edge-edges
• 1 edge-middle and 2 edge-edges make an "edge"
• 54 middle pieces: 6 middle-middles, 24 middle-edges, 24 middle-corners
• 1 middle-middle, 4 middle-edges, and 4 middle-corners make a "middle"
• 1 middle-middle and 2 middle-edges make a "middle-3-row"
• 1 middle-edge and 2 middle-corners make an "edge-3-row"
Corner
A cubelet at the intersection of three faces. May be referred to by the names of the three faces that meet there. For example, UFR and DBL are corners.
Direction
One of the three ways in which a face may be turned. + means a quarter-turn clockwise, - means a quarter-turn counterclockwise, and * means a half-turn.
Edge
The cubelets comprising the intersection of two faces. An edge on the 3x3x3 cube consists of 3 cubelets (and so on). An edge may be denoted by the names of the two intersecting faces; for example, UF and FR are edges. Confusingly, the term "edge" is sometimes used to refer to just the "middle" edge pieces -- all the edge cubelets except the corner cubelets on either end.
Edge cubelet
One of the cubelets on an edge, except for the ones on the corners. On the 3x3x3 cube, there is one edge cubelet per edge. On the 5x5x5 cube there are three.
Edge-three
On the 5x5x5 cube, the 3 cubelets on one edge other than the corner cubelets.
Edge-two
On the 4x4x4 cube, the 2 cubelets on one edge other than the corner cubelets.
Face
One of the six surfaces of the cube. The faces are called U (top), D (down), R (right), L (left), F (front), and B (back), relative to the orientation of the cube in your hand.
Flip
Turn a single cubelet in place. Usually used for edge cubelets. See also rotate. Do not confuse with swap.
Inverse
The operation that exactly reverses the moves involved in another operation. Take each turn backwards, and take all of the turns in reverse order. For example, the inverse of (R2- D R2 D* R2- D R2) is (R2- D- R2 D* R2- D- R2).
Location
Where a cubelet is located. Also position.
Middle
On the 3x3x3, the fixed cubelet in the center of a face. On the bigger cubes, a general term for the whole group of non-edge and non-corner cubelets on a face.
Move
A single turn of one or more slices of the cube together. The notation for a move consists of:
1. A letter representing the face.
2. An optional list of numbers representing the slices (usually just one for the 3x3x3 cube, but frequently more for bigger cubes). If omitted, it is assumed to be '1'.
3. An optional character representing the direction. If omitted, it is assumed to be '+'.
There are multiple valid notations for any one possible move. For example, the following notations (if the cube is 3x3x3) all mean "turn the front face clockwise a quarter turn": F, F+, F1, F1+, B3-.
Operation
A sequence of moves.
Orientation
The way a cubelet is situated in its location. Center face cubelets (odd-sized cubes only) have 4 different orientations. Corner cubelets have 3. Middle edge cubelets (odd-sized cubes only) have 2. All other cubelets have only one.
Position
Where a cubelet is located. Also location.
Rotate
Turn a single cubelet in place. See also flip. Do not confuse with swap.
Slice
Also layer or slab. Nine co-planar cubelets on the 3x3x3 cube; 16 on the 4x4x4; 25 on the 5x5x5. The NxNxN cube is N slices deep any way you look at it. The slices are numbered 1, 2, 3, etc., relative to a face.
Swap
Switch the locations of 2 or 3 cubelets. Do not confuse with rotate.

Tom Magliery
tom@magliery.com