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:

- 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.
- It covers re-orientations of the entire cube, such as "roll the cube toward you, so that the top face comes the front".
- 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.

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.

- 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"

- 3x3x3 cubelets
- 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:
- A letter representing the
**face**. - 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'. - An optional character representing the
**direction**. If omitted, it is assumed to be '+'.

- A letter representing the
- 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