Rubik's Cube - Solution and Wolstenholme Notation

Rubik's Cube is a famous puzzle cube invented by Ernö Rubik in 1974.

For those unfamiliar with the cube, the basic concept is that the cube is made up of 27 cubelets. The exposed faces of these each have a different colour. Rubik invented a mechanism whereby any layer of cubelets (i.e. 9 cubelets) can be rotated in a clockwise or an anticlockwise (counterclockwise) direction, independently of the other layers. By doing this many times at random, the colours displayed on the faces become jumbled up. The objective of the puzzle is to restore the initial position in which each side of the cube shows only one colour.

The approach to solving Rubik's Cube and the Wolstenholme notation and tools (sequences of twists) used are best seen in the Kublitz Cube application - an electronic version of the Cube. This is a version of the Cube in which you can effectively see, or know, the colours on all the cubelets without needing to turn the Cube around. This is effected by providing gaps between the 27 cubelets and by ensuring that the colours on any pair of opposite cubelet faces are identical; so, while you cannot see the colours on the back of the Cube, you can see those on the forward-facing sides of the back cubelets, which are the same.

You can find an online version of Kublitz Cube here. Tap on the Help button to see the approach to solving the Cube and the Wolstenholme notation (which is also given below). With the Kublitz Cube application you don't simply get the approach to solving the Cube, but a means of putting it into practice.

Wolstenholme Notation

The Wolstenholme notation is concerned with specifying rotations of the cube in order to solve it. In order to introduce it, we shall first give a brief outline of the Kublitz Cube referred to above, and show some images of it that also demonstrate the names of the outer faces or layers of the cube.

Outline of the Kublitz Cube

The graphics of the Kublitz Cube are intended to represent a large cube made up of 27 smaller cubes, or cubelets - hence the name Kublitz, a word that sounds the same. Three key features of the Kublitz Cube cubelets are:

  1. The cubelets are separated from adjacent cubelets by a significant gap;
  2. The colours of faces on opposite sides of every cubelet are always the same;
  3. The colours of the faces on the outside of the larger cube are one of six colours: red, pink, blue, green, white or yellow. The colours of the faces on the sides opposite to these on every cubelet are, from 2) above, also these colours. All other cubelet faces are grey.

The large cube (often referred to as just 'the cube') can also be considered as comprising 9 layers of 9 cubelets: a front layer and a back layer; a top layer and a bottom layer; a right layer and a left layer; and the middle layers lying between each of these pairs.

The large cube is designed so that it has a goal, or reset, state, in which the 9 cubelet faces of each layer of 9 cubelets are the same colour. This is equivalent to the state in which the colours displayed on the outside of every one of the 6 faces of the large cube are all the same colour (a different colour for each of the 6 faces), as shown below. This is effectively the same goal state as a traditional physical cube.

To demonstrate that the opposite faces of a cubelet are the same colour, if we rotate the cube 180 degrees around the top-bottom axis, it now appears as follows:

Note how the cubelets at the front, which were displaying red on their forward-looking faces, are now at the back, displaying red on their forward-facing faces - the faces that had been backward-facing when the cubelets were at the front of the large cube. Likewise, you can see that the cubelets on the right that were displaying blue on their right-facing sides are now on the left, but still displaying blue on their right-facing sides.

This design, and the view you get as shown above, means that you are able to know what colour is on every face of the large cube, since the opposite side of every hidden face, which has the same colour, is visible. This is a major difference, and improvement, on traditional physical puzzle cubes and their visual representations.

Basic Wolstenholme notation

In the images above, there are 12 arrows at the corners of the cube concerned with rotating the 6 outer layers.

Near each pair of arrows you will see a single letter. This is the way this notation refers to these layers:

The rotation of a given layer is specified in this notation by one of the following, placed after the layer letter:

So, LA means turns the Left layer 90 degrees anticlockwise (counterclockwise).

Important: Please note that the terms clockwise and anticlockwise are always used to indicate the direction of rotation when facing that particular side or layer from the outside of the large cube. This means that an anticlockwise rotation of the Back layer looks like a clockwise rotation when viewed from the front.

In addition, the notation uses a three-letter specification for cube rotations, in which the letter C is added to the end to signify that the entire cube is to be rotated. So, FOC means that the entire cube is to be rotated around the Front-Back axis in a clockwise direction when viewed from the Front. Note that the cube rotation FOC is the same as the cube rotation BAC.

When specifying sequences of rotations, we generally join two specifications of layer rotations together to form a 4-letter 'word'. So, ROTA specifies a clockwise rotation of the Right layer followed by an anticlockwise rotation of the Top layer.

These 4-letter 'words' and the 3-letter cube specifications often form recognizable words or names, or maybe sequences of letters that sound like words. Examples include: FOTO, ROC, BAC, ROTA, RITA, ROTI, or RIFA. This is the primary reason for choosing the notation above, since we humans are good at remembering words and can build stories around them to help us remember them.

Top-layer tools

The Kublitz Cube app includes four top-layer tools, which are sequences of rotations that have an overall impact on the Top layer, but not on the two layers below (see the app for more details). The sequences for each of the four tools are specified using the Wolstenholme notation and 'words', and have associated with them mnemonics that can help you remember them. You'll have to get used to thinking of eating rat, and know that a roti is a flat bread, often eaten as a wrap. These demonstrate the usefulness of having word-like rotation specifications.

Flipper tool
FOTO ROTA RAFA (photo rota rougher)
Flipper and his dolphin friends are forming a photo rota to have their pictures taken in turn, but the rougher sea may prevent this.
Swapper tool

LOTA RATO LATA ROTI (lotta rat-oh latter roti)
Lunch arrives: Cheese and a lotta rat! Oh, I'd rather swap the latter for a roti.

Twister tool
RATA ROTA RATI ROTI (ratter rota ratty roti)
The Ratter Rota shows the rat-catchers' work plans for the week, but with a twist, as it says who has to make the ratty roti for lunch.
Looper tool

RITA FOBA RIFA BOTA RI (Rita Fo B.A. reefer boater rye)
Loopy Rita Fo B.A. is celebrating her graduation smoking a reefer, wearing a boater and drinking rye whiskey.

These four top-layer tools can, of course, be used with a physical cube - and can you really say you've solved the cube until you've solved a physical cube? If you want to solve the cube without having instructions around, you'll need to remember the tool sequences, hence the mnemonics above.

Extended notation

The notation in Kublitz Cube was designed specifically for rotations of single outside layers of cubelets and of the entire cube. While all sequences can be specified using only this simple notation, more advanced puzzlers might wish to specify other moves more concisely. For example, they might want to refer to the rotation of the middle layer, between Front and Back, by 90 degrees clockwise as seen from the Front. Now this could be specified as three rotations using the simple notation: FOC FABO (i.e. rotate the complete cube clockwise then turn the Front and Back layers back again), but this is not concise.

The notation can be easily extended by references to the specific layers. In order to maintain the word-like notation, each layer reference begins with the letter E and is then followed by one or more of the following letters: N is the layer nearest to the face specified; M is the middle layer, i.e. the second layer back from the face specified, and X is the extreme layer furthest from that face. The E followed by letters denoting the layer or layers to be rotated are specified between the face letter and the rotation direction letter. So, the rotation of the middle layer, between Front and Back, by 90 degrees clockwise as seen from the Front can be specified as FEMO (or as BEMA). The rotation of the top two layers in an anticlockwise direction as seen from the Top can be specified as TENMA (or DEXMO). The rotation of the Right layer by 180 degrees could be referred to as RENI or LEXI, but you would normally use the simple RI for this, unless the extended notation helps you to remember the move.

David Wolstenholme