4x4 Parity: Fix Flipped Edges and Swap Two Corners

How many kinds of 4x4 parity are there?

Two, named after the step of the advanced 3x3 method where they show up:

• OLL parity: while making the yellow face, the number of top edge pairs with yellow facing up is 1 or 3. On a 3x3 it is always 0, 2 or 4 (Ruwix gives exactly this sign: an odd number of yellow edges facing up).
• PLL parity: at the last step, the top pieces sit in a way no 3x3 algorithm can solve. Most often just two edge pairs are swapped, or, if you place the edges before the corners, just two corners.

According to the Speedsolving.com Wiki, each kind turns up half the time and the two are independent. So out of four solves, on average one has no parity, one has only OLL parity, one has only PLL parity and one has both. Parity is an everyday thing for 4x4 solvers.

Why does a 4x4 have parity?

1. You start with (input)
3. After the formula

Each edge pair on a 4x4 is made of two small edge pieces (wings). Look at how they move when you turn:

• An outer turn such as R moves eight wings in two loops of four.
• An inner turn such as 2R (only the second layer from the right) moves exactly four wings in one loop: the edges FU, BU, BD and DF each lose one piece. The picture above shows this.

Mathematicians call one loop of four an odd permutation and two loops of four an even one. Flipping an edge pair in place is really swapping its two wings, which is odd too. On a 3x3 you only turn outer layers, so that odd state can never appear. On a 4x4 you turn inner layers many times while building the centres and pairing the edges, and nobody counts whether that was an even or odd number of times. If it was odd, the cube carries an edge pair "flipped on its own" into the 3x3 stage: that is OLL parity.

PLL parity comes from the centres. The four centre pieces of one colour look exactly alike, so they can swap places without you seeing it. That lets a 4x4 reach states where two edge pairs are swapped and everything else is solved, which is impossible on a 3x3. The Speedsolving.com Wiki gives a neat rule: every OLL parity algorithm has an odd number of inner quarter turns, and every PLL parity algorithm an even number. The two algorithms below follow it.

OLL parity: how to spot it

Once the first two layers are done and you're on the yellow cross, count the top edge pairs with yellow on top:

• 0, 2 or 4: normal, carry on with the yellow cross step.
• 1 or 3: OLL parity.

The picture above is an example with 3 edges up. The picture below has 1. The algorithm under each picture is OLL parity followed by the guide's yellow cross algorithm (the second one has an extra U to line the cube up).

This is the yellow cross step you know from the 3x3:

OLL parity: the algorithm

Turn U to bring any top edge pair to the edge between the top and front faces, then do:

2R2 B2 U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' B2 2R2

It flips only that edge pair. Everything else stays put, centres included. Afterwards the number of edge pairs with yellow up is always even, and you carry on with F R U R' U' F' as on a 3x3.

Memory tip: the algorithm starts with 2R2 B2 and ends with B2 2R2, like two brackets around the middle. The middle alternates a U2 (or F2) with an inner-layer turn. Practise it slowly in chunks rather than trying to remember the whole string at once.

It's long, so the Speedsolving.com Wiki also lists many shorter versions, or versions with wide turns for speedsolvers. They usually mix up a few more top-layer pieces, so beginners are better off with the one above first.

PLL parity: how to spot it

1. You start with (input)
3. After the formula

In the guide you put the yellow corners in place first (the Put the yellow corners in place step), then the edge pairs. So PLL parity always shows up at the very last step: all corners are right but the top edge pairs don't match any picture of the Put the yellow edges in place step. Three shapes are common:

• Two opposite edge pairs swapped (front and back): shown in the algorithm section below.
• Two adjacent edge pairs swapped: the picture above, front and left.
• All four edge pairs one place round in a loop: the picture below.

1. You start with (input)
3. After the formula

The fix is the same for all three: do the PLL parity algorithm once, and what's left is a normal last-step case. In the two pictures above, the part after the parity algorithm is the guide's step 7 algorithm, R U' R U R U R U' R' U' R2 (the top picture has an extra U before and after to line the cube up).

PLL parity: the algorithm

1. You start with (input)
3. After the formula

2R2 U2 2R2 Uw2 2R2 2U2

Uw2 turns the top two layers half a turn; 2U2 turns only the second layer from the top half a turn. The algorithm swaps the front and back top edge pairs and touches nothing else. The Speedsolving.com Wiki credits it to Chris Hardwick. If 2U2 feels awkward, use 2R2 U2 2R2 Uw2 2R2 Uw2 U2 instead; the result is exactly the same.

For two opposite edge pairs, turn U so they sit at the front and back, then do the algorithm and you're done. For the other two cases, just do the algorithm with the cube as it is, then finish the last step as on a 3x3.

Two corners swapped on a 4x4

If you solve the last layer in a different order from the guide, edges first and corners after, PLL parity can show up as just two corners swapped, with every other piece solved. That can't happen on a 3x3; on a 4x4 it can, and it is still PLL parity. The fix: do the PLL parity algorithm to swap two edge pairs as well, which turns the cube into a familiar 3x3 case, then do that 3x3 algorithm.

Two adjacent corners

1. You start with (input)
3. After the formula

1. Turn U so the two swapped corners are on the right: front-right and back-right (left picture).
2. Turn the whole cube y (like U, but holding the whole cube), do 2R2 U2 2R2 Uw2 2R2 Uw2 U2, then turn the whole cube y' back. This swaps the left and right edge pairs.
3. The cube is now a 3x3 T-perm case: do R U R' U' R' F R2 U' R' U' R U R' F' and you're done (right picture).

Two diagonal corners

1. You start with (input)
3. After the formula

1. Turn U so the swapped corners are front-left and back-right (left picture).
2. Same as above: y, then 2R2 U2 2R2 Uw2 2R2 Uw2 U2, then y'.
3. The cube is now a 3x3 N-perm case: do R U R' U R U R' F' R U R' U' R' F R2 U' R' U2 R U' R' and you're done (right picture).

If you'd rather not turn the whole cube, replace y, the algorithm, y' with 2B2 U2 2B2 Uw2 2B2 Uw2 U2: same result, using the inner back layer instead of the inner right layer.

Both parities: which one first?

OLL parity first, PLL parity second, because OLL parity shows up earlier:

1. At the yellow cross step, 1 or 3 edge pairs facing up: do 2R2 B2 U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' B2 2R2, then finish the cross.
2. Carry on with the yellow face and the yellow corners as in the guide.
3. At the last step, the edge pairs match no picture: do 2R2 U2 2R2 Uw2 2R2 2U2, then finish the last step as usual.

According to Ruwix, getting both costs a speedsolver about 6 to 7 extra seconds. Beginners lose much more, so practise these two algorithms on their own until your hands do them without thinking.

Not parity: three easy mix-ups

Before you reach for a parity algorithm, check these three things. They make the cube look "unsolvable", and no parity algorithm fixes them:

• Centres in the wrong colour order. For example red and orange swapped. With white on top and green in front, red must be on the right. Check against a corner piece.
• An edge pair built from the wrong pieces. Two wings that look alike but differ in one colour. Look closely at every pair before the 3x3 stage.
• First two layers not finished. Only count yellow edges once the two bottom layers are completely right.

Video

Want to see it moving? The short video below from the Noah Richardson channel on YouTube shows more of this. The video belongs to that channel and is embedded straight from YouTube.

Sources

• 4x4x4 parity algorithms – Speedsolving.com Wiki
• Parity on the 4x4 Rubik's Cube – Ruwix