Before we look at the algorithm to solve a 2 x 2 Rubix cube, it is important to understand the rotations of the faces.
A 2 X 2 Rubix cube has 8 pieces. Each of these pieces has a position in the cube. Unless a piece is at its final position and has the right orientation, the Rubix cube cannot be solved.
We being by forming a white square at the bottom. The idea is to get the first while square at the bottom, after which the remaining squares can easily be moved to complete the bottom layer.
The piece that is to be moved to the bottom-right corner is moved at top-right position.
To solve the bottom layer you could use the below algorithmic rotations.
Note : Below 4 rotations are not always needed to place the piece at the bottom-right corner. Stop the rotations once the piece gets correctly placed at the bottom-right corner.
R U R' U'
( Right ) -> ( Up ) -> ( Right Inverted ) -> ( Up Inverted )
Repeat Steps a) and b) to solve the bottom layer.
Check: If you don’t have same color tiles on the top layer, you may have
a) Not a single yellow tile on the top.
b) A single yellow tile on the top.
c) Diagonally placed yellow tiles on the top.
d) Yellow tiles in the same column / row on the top layer.
R U R' U R U U R'
( Right ) -> ( Up ) -> ( Right Inverted ) -> ( Up ) -> ( Right ) -> ( Up ) -> ( Up ) -> ( Right Inverted )
Repeat Step 2 till all the 4 tiles on the top layer have the same color.
Front / Back / Left / Right face having the same color tiles.
Keep the face that has the same colored tiles at the back and apply the algorithmic rotations once to solve the Rubix cube.
None of the 4 vertical faces ( Front / Back / Left / Right ) have the same color tiles
Apply the algorithmic rotations to get same color tiles on one of the 4 verticle faces. Keep this face at the back. Re-apply the algorithm.
Thus in this scenario the algorithm is used twice to solve the Rubix cube.
R' F R' ( B' X 2 ) R F' R ( B' X 2 ) ( R X 2 ) ( B' X 2 )
( Right Inverted ) -> ( Front ) -> ( Right Inverted ) -> ( Back Inverted ) -> ( Back Inverted ) -> ( Right ) -> ( Front Inverted ) -> ( Right ) -> ( Back Inverted ) -> ( Back Inverted ) -> ( Right ) -> ( Right ) -> ( Back Inverted ) -> ( Back Inverted )