How to solve a 2×2 cube
Solve your first layer, orient the four last-layer corners and then move those corners into their correct positions.
The three-stage solution
A 2×2 has only corner pieces, so the entire solve can be divided into three simple goals.
Solve one layer
Choose a colour and bring all four corners containing that colour into one layer. The four side stickers must also match: having four white stickers on one face is not enough if the surrounding colours do not form matching bars.
What the finished layer looks like
This example uses white, but you should gradually learn to start on any colour. Colour neutrality lets you choose whichever first layer is easiest on each scramble.
Start from a solved bar
If two corners are already solved relative to each other, hold that bar at the bottom-left. Preserve it while adding the other two corners.
Build from one corner
If there is no solved bar, pick one corner containing your chosen layer colour and find another corner that belongs beside it.
Make the first bar
In this example, the second white corner can be brought up with an L move—but first check whether it is the correct corner for that position. If the side colours do not match, move it to another position on the upper layer and try again.
Continue experimenting until two corners form a correctly solved bar, then place that bar at the bottom-left.
Insert the remaining corners
With the solved bar at the bottom-left, find the corner that belongs in the front-right position. Place it above the slot and repeat the four-move trigger until it inserts correctly:
R U R' U'Show move-by-move notation images
Once three corners are solved, keep the final empty slot at the front-right, find the last corner and use the same trigger.
Orient the last layer
Turn the cube over so the solved layer faces down. Count how many stickers of the opposite colour face upwards. With a white first layer, count the yellow stickers on top.
Every case below uses the same algorithm. Only the way you hold the cube changes.
R U R' U' R U2 R'Show move-by-move notation images
One facing up
Place the single upward-facing yellow sticker at the front-left, then perform the algorithm.
Two facing up
Turn the upper layer until the two upward-facing yellow stickers are on the right, then perform the algorithm.
None facing up
Find two yellow stickers on the sides that point in the same direction. Hold both of those stickers on the left, as shown, and perform the algorithm.
If the top is not yet completely yellow, identify the new case, adjust the holding angle and perform the same algorithm again.
Orientation complete
All four yellow stickers now face upwards. The side colours may still be mixed; that is expected. The last step moves the corners into their correct positions without changing the yellow face.
Permute the last-layer corners
Turn the upper layer up to four times and check whether two adjacent corners can form a fully solved side. The algorithm is the same whether or not you initially find a solved pair.
R U R' F' R U R' U' R' F R2 U' R'Show move-by-move notation images
Hold the solved corners on the left
If two corners are already correct relative to each other, hold that solved pair on the left and perform the algorithm once.
Perform the algorithm from any angle
If no two corners match after checking all four U-layer positions, perform the algorithm once from any angle. This creates a solved pair. Put that pair at the front-left and perform the algorithm again.