Solve a Rubik's Cube without algorithms: blocks, commutators

What is intuitive solving?

The beginner method gives you one algorithm for each situation. That's a quick way to start, but if you forget an algorithm, you're stuck.

Intuitive solving goes the other way. You only need a few ideas about how the pieces move, and you can use them in any situation. Ryan Heise, the author of the Rubik's Cube Solutions site, groups these ideas as fundamental techniques: align then join, treating a block as one piece, moving things out of the way, breaking and rebuilding, conjugates and commutators (Ryan Heise).

You don't have to give up algorithms completely. Even the sequences you come up with yourself are algorithms. The difference is that you know why they work, so you can adjust them when the situation is slightly different.

How to hold the cube and read the pictures

Hold the cube like this for the whole post:

• White on top, green at the front, red on the right.
• In the blockbuilding examples, we build a 2x2x2 block at the bottom front right corner. This block is made of the yellow, green and red corner, the three edges around it (yellow-green, yellow-red, green-red) and three centres.

Each picture shows two cubes. The cube on the left is the situation you have, and the cube on the right is the result after the sequence. Only the pieces being discussed and the centres are coloured. All the others are grey.

If you're not used to the notation R, U, F and the apostrophe yet, see Rubik's Cube notation for beginners.

Technique 1: blockbuilding

Blockbuilding means putting pieces together into small blocks, then joining those blocks in very few moves. According to the SpeedSolving Wiki, this is the intuitive way of solving used in steps of Petrus, Heise, Roux and ZZ, and it appears in almost every solution in the fewest moves event (FMC) (SpeedSolving Wiki).

The main idea: a finished block moves like one big piece. Ryan Heise describes this as treating a block as one piece (Ryan Heise). You don't need to worry about each piece inside the block. You only need to bring the whole block to the right spot.

Example: joining a square into place

The situation:

• Three pieces of the 2x2x2 block (the yellow, green and red corner, the yellow-red edge and the green-red edge) are already stuck together as a 2x2 square in the upper back half of the right face.
• The yellow-green edge is in the middle of the front face, on the right.

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

The solution takes only two moves:

1. F: brings the yellow-green edge down to its spot at the bottom of the front face. The square is at the back, so F doesn't touch it.
2. R2: turns the right face half a turn. The whole square travels down to the front like one piece and joins the edge you just placed. The 2x2x2 block is done.

If you placed the pieces one at a time, this would take far more moves. That's the power of blockbuilding.

Technique 2: align then join

Ryan Heise points out that a corner and an edge have many more ways of sitting wrong than right, so putting them together at random rarely works (Ryan Heise). His approach has two steps:

1. Align: turn things so the two parts you want to join are exactly one move apart, with their shared colours next to each other.
2. Join: make that move.

Example: lowering a pair into the block

The situation:

• The yellow-green edge and the yellow-red edge are already in place in the bottom layer.
• The yellow, green and red corner and the green-red edge are stuck together as one pair in the top layer, right above their spot.

If you do R straight away, the pair gets dragged along with it. So you need to align first. The solution is U R U' R':

1. U: moves the pair to the front left, off the right face.
2. R: lifts the yellow-red edge up to the middle of the front face, on the right.
3. U': brings the pair back. Now the pair and the yellow-red edge sit next to each other on the right face, in the same order as when solved. This is the align step.
4. R': joins them. The whole group goes down, and the 2x2x2 block is done.

If you've already followed the beginner guide to the 3x3 Rubik's Cube, this sequence is the first half of the algorithm for placing layer 2 edges. Now you know why it works.

Technique 3: move out of the way, then put back

Sometimes the move you need would break a block you've already finished. The fix: move that block somewhere else for a moment, make the move you need, then bring the block back. Ryan Heise describes these two ideas as moving out of the way and breaking and rebuilding (Ryan Heise).

Example: placing the last edge of the block

The situation:

• The yellow, green and red corner, the yellow-red edge and the green-red edge are already in place.
• The yellow-green edge is still in the top layer, on the right.

An F move would bring the edge down, but it would also drag the corner and the green-red edge along with it. The solution is R' F R:

1. R': turns the three finished pieces away from the front face, down to the lower back half of the right face. At the same time, the yellow-green edge comes down from the top layer to the middle of the front face, on the right.
2. F: brings the yellow-green edge down to its spot. The other three pieces are no longer on the front face, so they aren't touched.
3. R: returns the three pieces to where they were. The 2x2x2 block is done.

One R' move does two jobs: it clears the way and it puts the edge in position. Finding moves like this is how intuitive solvers save moves.

Technique 4: commutators

Near the end of a solve, every remaining piece sits among parts that are already solved. Blockbuilding isn't enough any more: you need to swap a few pieces without damaging the rest. That's exactly what a commutator does.

According to the SpeedSolving Wiki, a commutator is the sequence A B A' B': do A, do B, undo A, then undo B. If A and B don't affect each other, the whole sequence does nothing. Only the pieces in the overlap of A and B change, and every other piece goes back to where it was (SpeedSolving Wiki). Ryan Heise sums it up: make the overlap as small as possible, but not zero (Ryan Heise).

When the overlap is just one piece, the commutator swaps exactly three pieces in a circle. Cubers call this a 3-cycle.

Example: swapping three corners

Here A is R' D' R and B is U. A replaces exactly one piece in the top layer (the top front right corner). B only turns the top layer. Their commutator is R' D' R U R' D R U'.

The situation: three corners are in the wrong place, and every other piece is finished.

• At the top front right position is the white, blue and red corner.
• At the top back right position is the yellow, green and red corner.
• At the bottom front right position is the white, green and red corner.

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

Step by step:

1. R' D' R (A): brings the white, green and red corner up from the bottom to its spot on top. The white, blue and red corner gets pushed down into the bottom layer. The bottom layer is mixed up for now, but it will be put back.
2. U (B): turns the top layer so the yellow, green and red corner moves into the top front right position.
3. R' D R (undoing A): brings the yellow, green and red corner down to its spot, and at the same time brings the white, blue and red corner back up. The bottom layer returns to how it was.
4. U' (undoing B): turns the top layer back. All three corners are now in place and facing the right way.

Edges have their own commutators too, built on the same pattern. The SpeedSolving Wiki lists many examples for edges, for twisting corners and for flipping edges (SpeedSolving Wiki).

Conjugates: set up, do, undo

The commutator above only works when the three corners are in exactly those three positions. What if one corner is somewhere else? Use a conjugate.

Ryan Heise describes a conjugate as the sequence X Y X': X brings the pieces to the exact spots Y needs, you do Y, then X' takes back everything X did (Ryan Heise). The setup X is fully removed, so all that remains is the effect of Y on the pieces you wanted.

Example: swapping three corners in the top layer

The situation: three corners in the top layer are in the wrong place, and every other piece is finished.

• At the top front left position is the white, green and red corner.
• At the top front right position is the white, blue and red corner.
• At the top back right position is the white, green and orange corner.

The solution is L D L' R' D' R U R' D R U' L D' L':

1. L D L' (setup): brings the white, green and red corner from top front left down to bottom front right. The other two corners in the top layer aren't touched. Now the three corners are in exactly the three positions the commutator above deals with.
2. R' D' R U R' D R U' (commutator): swaps the three corners. The white, green and red corner and the white, blue and red corner go into place. The white, green and orange corner goes down to bottom front right.
3. L D' L' (undo): reverses the setup. The white, green and orange corner goes up to its spot at top front left, and the bottom layer returns to how it was.

The result: the three top layer corners are in place, and the rest of the cube is untouched. With conjugates and commutators, you can solve the very last pieces yourself without a table of algorithms.

Putting the four techniques into one solve

The Heise method is the clearest example of intuitive solving. Ryan Heise's site splits it into four steps (Ryan Heise):

1. Build four small square blocks. The blocks don't need to be the same colour.
2. Join the blocks together and orient the remaining edges.
3. Solve the remaining edges and two corners.
4. Solve the last three corners with a commutator.

The SpeedSolving Wiki credits Ryan Heise as the first person known to use blocks of mixed colours in speedsolving, in this very method (SpeedSolving Wiki).

If Heise still feels too hard, you can start with something gentler:

1. Solve with the beginner method, but replace the layer 1 and layer 2 steps with blockbuilding: a 2x2x2 block first, then grow it into two layers.
2. Keep your layer 3 algorithms as they are.
3. Once you're comfortable, slowly replace the layer 3 algorithms with commutators and conjugates like the examples above.

Practice exercises

1. Recreate each example. From a solved cube, do the example's moves in reverse to create the situation, then solve it again without looking at the post.
2. Build only a 2x2x2 block. Scramble the cube, then build just one 2x2x2 block at any corner. Count your moves, and next time try to find a shorter way. Once this feels smooth, move on to growing the block.
3. Find a commutator yourself. Pick an A that replaces only one piece in the top layer (for example R' D' R, or its mirror L D L'), then try it with U, U' or U2 as B. Watch which three pieces swap.
4. Say it out loud. Before each move, say what it does: align, join, move out of the way or put back.

An example for exercise 3: with U2 as B, the commutator R' D' R U2 R' D R U2 still swaps exactly three corners, but it uses the top back left corner instead of the top back right corner.

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

Common mistakes

• Forgetting to undo. Conjugates and commutators only keep your solved parts if you reverse exactly the moves you made.
• Too much overlap. If A replaces several pieces in the layer B turns, the commutator will mix up more than three pieces.
• Joining the wrong pair. Before joining, look again: stickers of the same colour must sit next to each other on the same face.
• Giving up too soon. Intuitive solving is slower than using algorithms at first. That's completely normal.

Next steps

Once you're comfortable with blockbuilding, you'll find methods like Petrus and Roux much easier to get the hang of, because they rest on this very idea. If you'd like to go the CFOP way, the post What is CFOP has a general route to follow.

With the EasyRubic app, you can scan your real cube with the camera and see the solution step by step on a 3D cube. Get EasyRubic to try the techniques from this post on your own cube.

Video to watch

Prefer to see it in motion? The short video below, from the MD Ăn Bốc channel on YouTube, adds more pictures to what this post covers. The video belongs to that channel and is embedded directly from YouTube.

Sources

• Blockbuilding (SpeedSolving.com Wiki)
• Commutator (SpeedSolving.com Wiki)
• Rubik's Cube Solutions – Ryan Heise
• Align then join – Ryan Heise
• Conjugates – Ryan Heise
• Commutators – Ryan Heise
• Heise method – Ryan Heise