First layer

A first layer whose corners are in place, one of them possibly twisted.
The 2x2 has only corners. Every method starts with one side: a face (four stickers of one colour) or a layer (a face with its side colours matching). Then:
  • •
    Ortega: a face, then orient the opposite face (7 OLLs), then permute both layers at once (6 PBLs): 13 algorithms.
  • •
    CLL: a full first layer, then one algorithm for the whole last layer (40). Fewer looks; the first layer costs a few more moves.
  • •
    EG: a face only, then one algorithm that also fixes the bottom's permutation: CLL (bottom solved), EG-1 (adjacent swap below) and EG-2 (diagonal swap below), 120 algorithms. The top method for competition.
  • •
    TCLL: as CLL, but the first layer may finish with one corner twisted in place; TCLL+ and TCLL- solve that twist with the last layer. Usually learnt after EG, for the layers where the twist saves moves.
A first face done; the top is an Ortega OLL case (Antisune).
Checked by the engine: Ortega OLL Anti Sune
R U R' U R U2 R'
Both faces done: one PBL left (an adjacent swap on top).
Checked by the engine: PBL Adj
R U R' U' R' F R2 U' R' U' R U R' F'

Ortega OLL

7 cases · Orient the 2x2's top face over a solved bottom face: 7 cases.

PBL

6 cases · Permute both layers of a 2x2 at once: 6 cases.

CLL

40 cases · Solve the 2x2's last layer in one algorithm after the first layer: 40 cases.

EG-1

40 cases · Solve a 2x2 whose first face has an adjacent corner swap underneath: 40 cases.

EG-2

40 cases · Solve a 2x2 whose first face has a diagonal corner swap underneath: 40 cases.

TCLL+

43 cases · Untwist a clockwise-twisted first-layer corner and solve the last layer: 43 cases.

TCLL-

43 cases · Untwist an anticlockwise-twisted first-layer corner and solve the last layer: 43 cases.

Beginner layer by layer

A first 2x2 solve needs only two algorithms. Build a face, then turn the layer until each of its sides is one colour; that is the first layer. Hold it on the bottom and repeat the Sune (R U R' U R U2 R') until the top is one colour, turning the top between tries so a corner whose top colour faces you sits at the front left. Then swap the corners that are out of place with the corner swap of the 3x3's beginner method, held with a matching pair at the back.