How to Solve the Rubik's Cube
There are several notations; please refer to this notation guide.
- 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.
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 (demo: scramble)
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′ y2 F D2 F2 R F R′ B′ D F D′ B D F′ F2 D M D2 M′ D F2 (54 moves, demo)
While the above method may be good for a beginner, it is too slow to be used in speedcubing. The most popular method for speedcubers is very similar to the 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
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. 
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. 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.  ZBLL algorithms can be found on Doug Li's webpage. 
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.   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  and PLL , Orientation of LL and Permutation of LL, and COLL  and EPLL , 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 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. 
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. 
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. 
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. 
Corners first methods
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. 
Jelinek Method: Created by Josef Jelinek. This method is very similar to Waterman's. 
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).
Three "Levels of Difficulty"
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".
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.
Other solution pages
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.
- Rubik's Cube solution for Beginners (rubiksplace.com)
- Beginners Solution By Michiel van der Blonk
- Beginners solution By Christophe Goudey.
- Rubik's cube solution with images by Diego de Pereda
- 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 by Jasmine Lee
- Beginner's solution (unknown author, requires purchase for the later steps)
- Video tutorial by Tyson Mao.
- Blogspot Site
- How to solve a Rubik's Cube at the official Rubik's Cube website
- Beginners Solution (text) by Mark Jeays
- 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.
- Automatic cube solver
- Cube solver with 3D Java applet
- 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
- Rubik Puzzles describes other puzzles designed by Erno Rubik
- McNaughton, D. (November 1989 - February 1990). "The Rubik Cube: A three-stage approach to mastering it". Junior News (Al-Nisr, Dubai, UAE). http://dlmcn.com/rubik2.html.