2x2x2 Speedsolving Methods
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Although the 2x2x2 cube can be considered as just the corners of the 3x3x3 cube, there are several ways of solving it, ranging from simple to advanced. The best site for many of these methods, containing algorithms and fuller explanations, is Erik's 2x2x2 tutorial page, which actually contains all of these methods except for Stern-Sun and Lukasz's method. For any method, it helps to be color-neutral, so you can avoid bad cases.
The best 2x2x2 solvers can solve the cube in a global average of under 4 seconds.
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3x3x3-like Method
One of the easiest ways to solve the 2x2x2 is just like the 3x3x3. First, you insert the first layer corners, as in a 3x3x3 beginner's method, then orient the last layer, using an algorithm from 2-look OLL, pretending there is a cross on top, and finally do a PLL, switching two corners along with the two edges, or a three cycle of corners, if applies. Note that the 2x2x2 has a much higher chance of an OLL or PLL skip, as it has no edges. You can also sometimes use more efficient sequences, such as FRUR'U'F' for the "superman" orientation case.
LBL
This is an extension of the 3x3x3 method above. The first step is to solve one layer of the cube; this is an intuitive step, so you should try to figure out how to do it all during inspection, and do it in as few moves as you can. Then you can solve the orientation and permutation in two steps. If you have the time, you can even do the last layer in one algorithm by learning CLL, which has 42 algorithms. World record holder Edouard Chambon uses CLL algs.
Ortega
After a 3x3x3 method, the next step for most people is the Ortega method. This method is popular among Japanese cubers. First, solve one face intuitively; don't worry about solving a layer, because the face will be permuted later. Second, orient the opposite face, using the same OLL algorithms as on 3x3x3 (or more efficient ones if you want). Finally you permute both layers at the same time. The last step sounds difficult but there are only 5 possible cases, so it is quick to learn. In total, there are 12 algorithms to learn (11 without reflections).
For the first face, without color neutrality, the average move count in HTM is a surprisingly low 3.97, and no cases require more than 5 turns.
Guimond
This method is very popular because it is efficient. For the first step, solve just 3/4 of a face of opposite colors, which means you have to get three stickers of one color and the opposite color on one face. You can usually get this step in one move or less, so you can look forward to the next step. In the next step, you finish the face and orient the opposite one, so that you have two opposite faces of opposite colors. Then you sort the opposite colors by putting all of one color onto one face and all of the other color onto the other, and finally you permute both layers like in Ortega. Except for the last step, all of these steps usually take four or less moves, and experienced users can often look all the way through to the last step during inspection. Guimond requires 21 algorithms (or 13 without reflections).
Very Advanced Methods
There are a couple of very advanced methods for solving the 2x2x2, and only a handful of people know each one because they usually have over a hundred algorithms to learn. Don't even consider learning these unless you REALLY love the 2x2x2.
The EG method (Erik-Gunnar) only has two steps: solve one face, and then finish the entire cube in one algorithm. Without reflections, it requires 120 non-PBL algorithms.
The OFOTA method (opposite face orient, orient, all) has three steps. First you create an entire face of opposite colors, next you orient the opposite face while sorting the colors so you have two solid-color faces (like the second and third steps of Guimond), and finally you permute both layers at once. According to Erik's page, there are 87 non-PBL algorithms to learn, including reflections.
The SS method (Stern-Sun) has three steps. First you solve 3/4 of a face, then you finish the face while orienting the opposite face, and finally you permute both layers at once. Without counting reflections, there are 60 algorithms, including the 5 PBLs.
