Permuting Last Layer Corners: Position the Yellow Corners

Understanding Corner Permutation

At this point in the beginner method, you have a complete yellow face on top — all four corners and four edges have yellow facing up. However, while the orientation is correct, the corners may not be in their correct positions. A corner is in the correct position when its three colors match the three adjacent face centers, even if the corner's yellow sticker was already oriented correctly in the previous step.

To check whether a corner is in the correct position, look at its three colors and compare them to the three surrounding centers. For example, the corner between the red, blue, and yellow faces should have red, blue, and yellow stickers. If it does, that corner is positioned correctly. If the colors don't match, the corner needs to be permuted (moved to its correct slot).

Checking All Four Corners

Rotate the U layer and check all four corners by looking at each one individually. You'll find one of three situations:

All four correct: Lucky! Skip this step entirely and move to edge permutation.
One corner correct: This is the most common scenario. One corner matches its position, and the other three need to be cycled.
Zero corners correct: No corners are in position. Apply the algorithm once (with any corner at front-right), and this will move to either all-correct or one-correct state.

It's mathematically impossible to have exactly two or three corners correct — it's always zero, one, or four. This is a consequence of the cube's parity constraints.

The Corner Permutation Algorithm

The algorithm used to cycle three corners is:

U R U' L' U R' U' L

This 8-move sequence performs a three-corner cycle: it rotates three corners around while leaving the fourth corner fixed. The fixed corner is the one at the front-right position (FRU).

How to Apply It

Rotate the U layer until the one correctly-positioned corner is at the front-right (FRU) position.
Apply the algorithm: U R U' L' U R' U' L
Check if all corners are now correct. If not, apply the algorithm again (keep the same corner at FRU).
At most, you'll need to apply it twice.

If no corners are correct initially, apply the algorithm once with any corner at FRU. After the first application, exactly one corner will be correct. Rotate U to place that corner at FRU and apply the algorithm again.

Verifying Corner Positions

A common mistake is misjudging whether a corner is correctly positioned. Remember: you're checking position, not just the color facing up. A corner might have yellow facing up (correct orientation from the previous step) but be in the wrong slot. Look at all three stickers of each corner and verify they match the surrounding centers.

Tip: When checking corners, focus on the two non-yellow stickers. If those two colors match two adjacent centers, the corner is in the right position. This is faster than checking all three stickers.

The Mathematics Behind Corner Cycling

The U R U' L' U R' U' L algorithm is what mathematicians call a "3-cycle" or "three-element permutation." It takes corner A to position B, corner B to position C, and corner C to position A, while leaving corner D untouched. In group theory, any permutation of four elements can be decomposed into 3-cycles, which is why this single algorithm is sufficient to solve any corner arrangement.

Interestingly, you can never have a situation where only two corners need to be swapped (a "2-cycle" or transposition) without also requiring an edge swap. This constraint is built into the cube's mathematical structure and is related to the concept of parity — a deep topic that becomes relevant when solving larger cubes like the 4x4.

Moving to the Final Step

Once all four corners are correctly positioned, you're ready for the final step: permuting the last-layer edges. You're just one algorithm away from a solved cube! The edge permutation step uses a different algorithm to cycle three edges into their correct positions, and after that, your Rubik's Cube will be completely solved.