How to Solve the Rubik's Cube

## Notation

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There are several notations; please refer to thisnotation guide.

Briefly:

- There are six sides to the cube, presented as Front, Back, Left, Right, Up and Down. They are usually referred to by their one-letter abbreviations.
- In the isometric diagrams below, where a corner points out at you, you see the F, R and U sides. The F faces to the left.
- Movements are presented as one quarter rotation (90 degrees) of an external face per movement. This means that the center tile colors are not changed. In our diagrams, F is blue, R is red and U is yellow. The other three colors are typically orange opposite red, green opposite blue and white opposite yellow
- Quarter-rotations of that face's layer default to clockwise. Counter-clockwise rotations are often referred to as "inverted" and indicated by ′, for instance, R′. (The ′ is commonly read as "prime", "apostrophe", "tick mark", "anti-clockwise", "anti" or "i" for inverted). Half-rotations (180 degrees) are indicated by the digit "2", for instance, R2 (meaning 2 quarter-rotations following the one-letter abbreviation).
- To see what is happening on the sides of the other three colors, rotate the cube as a whole, described as rotating along the
*x*,*y*,*z*space axis, all pointing out of the page.*x*is R,*y*is U and*z*is F, but since this sort of move also changes the colors of the center-tiles, it is used sparingly.

B | D | F | L | R | U |

## Example solve

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As an example, let's consider a complete solve. 25 move scrambles are used to mix up the cube. Our sample scramble is:

- U B′ R2 D′ U′ R U2 B R′ B2 L2 R F2 R2 U2 R B U2 F2 L2 F2 D R B2 R2

The solve is:

- R′ B R D2 F2 L U′ F U R′ D R F D F′ F′ D′ F U2 R′ D′ R U2 U′ F′ D′ F U U B′ D′ B U′
*y*2 F D2 F2 R F R′ B′ D F D′ B D F′ F2 D M D2 M′ D F2 (54 moves)

## Methods

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There are a large variety of methods to solve the Rubik's Cube. Here we list methods that are described in some detail in this Wikibook, before briefly reviewing other methods in the following sections.

- Beginner
- Beginner (alternative)
- Fridrich Method

## Classical Methods

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The first widely publicized solutions to the Rubik's Cube appeared in the early 1980s, when quite a number of solutions where published in books and articles. For example, see Philip Marshall's comparison of various classical methods. Here we mention two solutions from around 1981 byDavid Singmaster^{[1]},and James Nourse^{[2]}.

One of the key observations was that the solution can be broken down into several steps. Most of the "standard" classical approaches solve the cube layer by layer. For example, solve1. all edges in the top layer, 2. the corners in the top layer, 3. the edges in the middle or horizontal layer, 4. the edges in the bottom layer, and 5. the corners in the bottom layer,which completes the solution. There are a number of relevant variations, for example step 2 and 3 can be combined (see the Fridrich method below), or step 5 is often split into first putting the pieces in the right place and then fixing their orientation. The bottom line is that such steps simplify the solution process because there exist algorithms (sequences of face turns) that can handle the individual steps efficiently. Solution methods differ not only in the steps, but also in the set of algorithms suggested for the individual steps.

As Marshall discusses, many of the more recent methods have their roots in these early solutions. This is also apparent in several of the methods that are listed below, which however include a number of improvements. The solution steps have been modified, and the sets of algorithms have been improved and optimized for different solutions steps. Furthermore, some of the early methods were not fully explained (or had gaps), and many improvements in presentation were made.

In the following, we comment on beginner methods, where the goal is simplicity (usually at the cost of efficiency), and on advanced methods, which give faster and/or shorter solutions (usually at the cost of increased complexity).

## Beginner Methods

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The term *beginner method* is used differently by different authors. A beginner method should be simple, but what is considered simple depends on the person, and changes quickly as experience is gained. As Marshall discusses, some of the early simple methods required 10 to 20 algorithms and needed 100 to 150 moves to solve the cube. For example, he reports an average of 110 moves for Nourse's method with 12 algorithms, which comes down to 100 moves adding shortcuts involving 20 algorithms in total.

A modern method could be called simple if it only needs 5 algorithms or less. If you are used to advanced methods with 50 plus algorithms, then 10 or less is also simple. Also, the algorithms should not be too long and complicated themselves. Some of the recent beginner methods with 5 or less algorithms are surprisingly efficient, requiring less than 100 and in some cases only about 70 moves. This is quite nice compared to some advanced speedcubing methods that require 40 to 60 moves or so on average but use 50 plus algorithms.

### Recent beginner methods

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**Solution guide at Rubiks.com**: The guide at the official Rubik's site seems to be one of the classic layer-by-layer methods. It lists 14 algorithms. [1]

**Heise's beginner method**: This is a representative example of an optimized, classical layer-by-layer method. The original strategy is credited to David Singmaster. It requires four algorithms. (Some of the easy steps are labeled intuitive and not counted as algorithms, but this is common practice.) The solution is presented with the help of animations. [2]

**Wikibook Beginner method**: Presented on these wikibook pages, this layer-by-layer method uses 5 to 8 algorithms, depending on how you count them. Although perhaps not optimal in the number of algorithms, it demonstrates a rather successful idea. Only 3 of 4 corner pieces and 3 of 4 edge pieces are solved in the first two layers. Leaving those pieces as free workspace can simplify some of the later steps. See the Petrus method below for the classical example of a block building method that avoids solving certain pieces too early.

**8355 method**: Another layer-by-layer method using two pieces as workspace. Requires 3 algorithms. In this case the workspace allows a simplification compared to, say, Heise's beginner method.[3]

**Philip Marshall's method**: An edges-first method that relies on only 2 algorithms. Reported to require only 65 moves on average by Marshall. Notice that the above layer-by-layer with workspace methods can be turned into edges-first methods if solving the three corners of the first layers is delayed until all edges have been solved. The Marshall method achieves its simplicity by using one particular 4-turn algorithm to do all the edges, and one 8-turn algorithm for the corners. Some elementary setup turns are needed as well. [4]

**Single algorithm methods**: The 8355 method and Marshall method can be reduced to single algorithm methods, see the Single Alg Cube Solution and the Y-move method. Basically, there are two elementary 4-turn commutators, the "S-move" and the "Y-move", which are used in these and other methods to solve the edges. It turns out that these commutators can also be applied repeatedly to replace the corner algorithms in the 8355 and Marshall method. There is a small loss of efficiency, but in this way single algorithm methods are possible. Another single algorithm method was originally developed by Camilo Vladimir de Lima Amaral[5], who called it "Less is more method" or "Amaral Method" [6].

Are there zero algorithm methods? The answer is yes since single face turns are not counted as algorithms, and the cube can certainly be solved with those. However, the idea of algorithms is to group single face turns into something manageable by humans. Relying only on "intuition" and single face turns has not led to a beginners method yet.

In conclusion, there are several recent beginner methods that improve on the classical layer-by-layer beginner's methods, although simple can mean different things. Simple methods with just 1 to 4 algorithms are possible, where the main point is that this is less than 10 or 20. Still, a beginner may prefer a printed list of 5 or 10 algorithms over a less explicit method using only a 1 or 2 algorithms. On the other hand, a method with fewer algorithms is easier to memorize and easier to understand than a more complex method.

"Algorithms of last layer":Some algorithms of fixing last layer-

1. Making cross-(F R U R' U' F' if you have a horizontal bar & F U R U' R' F' if you have a back-left hook)

2. Matching edge colours-(R U R' U R U2 R')

3. Fixing corners-(U R U' L' U R' U' L)

4. Matching corners-(D R' D' R)

## Faster methods

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While the above methods may be good for a beginner, they are too slow to be used in speedcubing. The most popular method for speedcubers is very similar to the Wikibook Beginner method above, except steps 2 and 3 are combined, and the last layer is solved in two steps instead of three. The inventor of this common method is Jessica Fridrich. With this method, speedcubers with good dexterity and memory can average under 20 seconds after a few months of hard practice. However, to learn the method you must learn 78 algorithms. There are methods just as fast that require far fewer algorithms to be memorized. Here is a brief synopsis of several popular speedcubing methods:

### Layer by Layer methods

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**Fridrich Method:** A very fast First 2 Layers (or F2L) method, start by solving a cross on one face, then proceeding to solve the First 2 Layers pairing up edge and corner combinations and putting them into their *slot*. This is followed by solving the Last Layer in two steps, first orienting all pieces (one color on the last layer), then permuting them (solving the ring around the last layer). The basic method has 78 algorithms (without the inverse of them), and is recognized as one of the fastest methods currently in use. [7]

**F2L Alternatives:** Methods that follow the same principle as Fridrich's method, but using different algorithms. Many of the algorithms are shared but there are a few differences, so there should be one to suit your fingers:

**ZB method:** This method was developed independently by Ron van Bruchem and Zbigniew Zborowski in 2003. After solving the cross and three c/e pairs, the final F2L pair is solved while orienting LL edges. This is known as ZBF2L or ZBLS. The last layer can then be solved in one algorithm, known as ZBLL. The ultimate method requires several hundred algorithms. Lars Vandenbergh's site has ZBF2L algorithms, used in his VH system. [10] There is also a more recent Google spreadsheet at [11]. ZBLL algorithms can be found on Doug Li's webpage. [12]

**ZZ method:** This method was created in 2006 by Zbigniew Zborowski, the co-creator of the ZB method. It has three basic steps: EOLine, F2L, and LL. [13] [14] EOLine stands for Edge Orientation Line. The orientation of edges is defined as either good or bad. Good meaning the edge can be placed into the correct position with a combination of R, L, U, D, F2, or B2, moves. Bad meaning it would require an F, F′, B, or B′ move to be moved into its correct position. Any F, F′, B, or B′ move will cause the four edges on that slice to change from its current state, good or bad, to the opposite state. The Line portion of EOLine is forming a line on the bottom of the cube that consists of the DB edge and the DF edge in their correct positions. The next step is F2L, First 2 Layers. It uses block building techniques to solve the two remaining 1x2x3 blocks of the F2L using only R, U, and L moves. This allows for very quick solving of F2L as it does not require cube rotation. The final step of the ZZ method is LL, Last Layer, and it can be broken into multiple steps or maintained as one depending on the algorithms used. There are two main approaches to this method OLL [15] and PLL [16], Orientation of LL and Permutation of LL, and COLL [17] and EPLL [18], Corner OLL and Edge PLL. The first, OLL and PLL, is to use one of 7 algorithms to solve the top layer (OLL) and then permute the edge and corners into their correct positions (PLL), this requires 21 algorithms. The total algorithms required for the first approach of solving LL is 28. The second approach to solving LL is to solve the top and the corners in one algorithm (COLL) and then solve the edges (EPLL). COLL requires 40 algorithms and EPLL requires 4, making the total 44 algorithms. The second approach is faster due to the ease of recognition and speed of execution of EPLL.

**VH method:** Created by Lars Vandenbergh and Dan Harris, as a stepping stone from Fridrich to ZB. First, F2L without one c/e-pair is solved with Fridrich or some other method. Then the last pair is paired up, but not inserted. Then it's inserted to F2L and LL edges are oriented in one go. Then, using COLL, corners of LL are solved while preserving edge orientation. Then edges are permuted. [19]

### Block methods

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**Petrus System:** Created by Lars Petrus. One of the shortest methods in terms of face turns per solve, the Petrus method is often used in fewest moves contests. Petrus reasoned that as you construct layers, further organization of the cube's remaining pieces is restricted by what you have already done. For a layer-based solution to continue after constructing the first layer, the solved portion of the cube would have to be temporarily disassembled while the desired moves were made, then reassembled afterwards. Petrus sought to get around this quagmire by solving the cube outwards from one corner, leaving him with unrestricted movement on several sides of the cube as he progressed. There are not as many algorithms to learn compared to the other F2L methods, but it takes a lot of dedication to master. The basis of the method is to create a 2 × 2 × 3 block on the cube, then proceed to solve a 3 × 3 × 2 block, but also flipping the edges on the Last Layer. Then the Last Layer is solved in two steps, first corners and then edges. [20]

**Heise method:** Created by Ryan Heise. First, one inner square and three outer squares are built intuitively. Then they are placed correctly while orienting remaining edges. After that you create two c/e-pairs, and solve the remaining edges. The last 3 corners are solved using a commutator. [21]

**Gilles Roux Method:** Another unique method, but works in blocks like the Petrus method. You start by solving a 1 × 2 × 3 block and then solve another 1 × 2 × 3 block on the other side of the cube. Next you solve the last 4 corners and finally the edges and centers. Has only 24 algorithms to learn. [22]

### Corners first methods

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**Waterman Method:** Created by Mark Waterman. Advanced corners first method, with about 90 algorithms to learn. Solve a face on L, do the corners on R and then solve the edges. An extremely fast method. [23]

**Jelinek Method:** Created by Josef Jelinek. This method is very similar to Waterman's. [24]

Create a solved 2 × 2 × 2 cube on one corner and rotate the remaining blocks (it may take a while but you will eventually solve it).

### Catta Method

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From its creator the Italian Giuseppe Catta, unique and different from all, based on structural pieces (in the case of the 3x3x3 cube of only 2 types, edges and medians), from the edges towards the center and hinged on the two opposite faces . Very important note as these 3 principles with the Catta method subsequently guide the resolution of Rubik's Cubes of any size NxNxN, except for implementing additional algorithms consistent respectively with two additional structural pieces with the 4x4x4 (4 "centres" instead of 1 "diagonal internal crowns", not piece, of the 3x3x3; and 2 lateral medians instead of 1 of the 3x3x3) and a further structural piece with the 6x6x6 ("symmetrical median internal crowns").

This method - discovered and codified not by chance by an architect - is extremely didactic, simpler, more orderly, understandable and easier to memorize than the others, and its approach is paradigmatic of a way of dealing with things as a whole and as a whole , to system, from general to particular. It teaches us to see the whole whole at once, as a system of parts rather than as a sum or juxtaposition of parts.

## Three "Levels of Difficulty"

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A procedure has been developed whereby a complete beginner can learn and master the Cube by moving up through three self-contained "Levels of Difficulty".^{[3]}

The lowest level deliberately maintains configurations in which every face exhibits horizontal and vertical symmetry, so it also enables a number of "pretty patterns" to be constructed - such as Chequers, Crosses, Stripes and Central 'Dots'. Level Two involves solving cubes which have been scrambled using only 180-degree turns. Techniques acquired in those earlier stages, remain useful when continuing up to the next level.

## How to solve the 2x2x2 (Mini/Pocket Cube)

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If you know how to solve the 3x3x3 cube and 4x4x4 (see above), solving the 2x2x2 can be achieved by treating the cube as a 3x3x3 in which the centre and edge cubes are in a permanent solved state regardless of what moves you make. In other words, the solution consists of only the corner-solving steps of the 3x3x3 solution. The important thing is simply to remember which side is which (since you can't see the middle layers, because they aren't actually there), although this shouldn't be too difficult.

## Other solution pages

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Here are some of the more popular solution pages listed. All are different, although they mostly use a similar layer by layer method. Usually you will need Java to see the animations used.

Using animations:

- Rubik's Cube solution for Beginners (rubiksplace.com)
- Beginners Solution By Michiel van der Blonk
- Beginners solution By Christophe Goudey.

Using pictures:

- Rubik's Cube solution with animations (rubiksplace.com)
- Simple solution by Rick Rayner
- Beginners' solution by Alan Chang
- Beginner's Guide (unknown author)
- Beginners' solution (translated into multiple languages) by Jasmine Lee
- Beginners' solution (unknown author, requires purchase for the later steps)
- Simplest method / Beginners' solution (How to Solve Rubix Cube)

Using video:

- Video tutorial by Tyson Mao.
- Blogspot Site
- How to solve a Rubik's Cube at the official Rubik's Cube website

Text only:

- Beginners Solution (text) by Mark Jeays

### Alternative methods

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- Solving the Rubik's Cube for Speed, a block method by Lars Petrus
- Solving the Rubik's Cube A corners first method by Matthew Monroe
- Ultimate solution to the Rubik's Cube An edges-first method by Philip Marshall, requiring the memorization of only 2 algorithms and requiring an average of only 65 moves to solve.
- The Single Alg Cube Solution and the Y-Move Method are beginner methods that each rely on only a single, four-turn algorithm.

### Solving Programs

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- Rubik's Cube Solver App
- Automatic cube solver
- Cube Explorer 4.10 – a fast program for finding optimal or near optimal solutions to the cube (less than 20 moves total!).

### Background on the mathematics

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- Rubik Puzzles describes other puzzles designed by Erno Rubik

## References

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- ↑ Singmaster, David (1981).
*Notes on Rubik's Magic Cube*. Harmondsworth, Eng: Penguin Books. ISBN0-907395-00-7. - ↑ Nourse, James G. (1981).
*The Simple Solution to Rubik's Cube*. New York: Bantam. ISBN0-553-14017-5. - ↑ McNaughton, D. (November 1989 – February 1990). "The Rubik Cube: A three-stage approach to mastering it".
*Junior News*. Al-Nisr, Dubai, UAE.`{{cite journal}}`

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