Solve a 3x3x3 cube

This page describes a method of solving a 3x3x3 cube.

This solution requires freely re-orienting the cube during the solve. This is unlike the typical beginner's method which holds the cube in a fixed orientation.

• Introduction
• Solve the cube
• Position Corners
• Position Edges
• Orient Edges
• Orient Corners
• Summary

Introduction

Many methods have been devised to solve a 3x3x3 cube. Methods have been designed for speed, fewest moves, blindfolded, one handed, and more. The intent of this method is to provide an intuitive solution to solving the cube. When given a scrambled cube, most people are able to solve a single face without any help. They do so by observing how the cube is affected when a face is turned and then applying what they learn in order to complete a face. It is hoped that this solution will provide insight so that the whole cube can be solved in a similar manner.

The typical beginner's method for solving a cube has several move-sequences (a.k.a. algorithms in cubing terminology) that must be memorized. The move-sequences are relatively short, but there is no apparent logic for why the move-sequences were chosen or how they work. This results in a situation where the cube is directing the solve and the human is just a Mechanical Turk that must carry out a series of seemingly arbitrary face-turns. If the human executes the face-turns correctly, the cube will be moved into a new state which will dictate a different series of seemingly arbitrary face-turns that must be used. This will continue until the cube is solved. To some degree, this process is true for every method of solving the cube, including this one. The difference is that this method provides a 'tool' for manipulating the cube in a useful manner. It is up to the human to apply the 'tool' in various ways to solve the cube. This allow the human to tell the cube what to do, rather than the cube telling the human what to do.

As with every solution, there are advantages and disadvantages.

Advantages:

• Understandable. The goal is to solve the cube in a reasoned manner.
• Buildable: New move-sequences build on earlier move-sequences.
• Conjugate-free: 'Setup' and 'teardown' moves are not required, but can be used.
• Eager: The hard work is done first, which makes solving later stages almost trivial.
• Mono-move: A single move-sequence is used. No left, right, or mirrored move-sequences are needed.
• Forgiving: The cube can often be returned to its starting state simply by repeating a move-sequence.
• One-handed: The solution can be performed using one hand without too much difficulty.

Disadvantages:

• Slow: A solve takes about eight minutes.
• Repetitive. The solution uses the same four-turn move-sequence over, and over, and over.
• Topless: There is no fixed top. The entire cube must be re-oriented multiple times during the solve.
• Speckled: Edge-cubies and corner-cubies are positioned without orienting them. This results in a jumble of colors which makes it difficult to locate cubies or identify the state of the cube.

In most examples, especially in the beginning stages, cubies are shown correctly oriented in their home positions. This is only done for the sake of clarity. When solving a real cube the cubies will almost never be oriented as depicted.

Many examples animate turning the layers of a cube. In all animations, the act of re-orienting the cube is never shown. Instead, the cube is held stationary and different faces are turned as if the cube were re-oriented. When solving a real cube, it is usually easier to physically re-orient the cube and turn the Front and Right faces, rather than hold the cube stationary and turn different faces. Either method will work.

If you are unfamiliar with the cube, it may help to learn about it.

The cube will be solved in four steps:

1. Move all eight corner-cubies to their home positions, ignoring their orientation.
2. Move all twelve edge-cubies to their home positions, ignoring their orientation.
3. Orient all twelve edge-cubies.
4. Orient all eight corner-cubies.

A 'tool' will be used to help solve the cube. The 'tool' is a four-turn sequence named HedgeSlammer. For this guide, the HedgeSlammer should be thought of as a single, indivisible move. Once a HedgeSlammer has begun, nothing else should happen until the HedgeSlammer is complete.

When solving a cube, if a wrong face is turned, or if a face is turned the wrong direction, there are two ways to correct the error. One way to correct the error is to un-turn the face. The other way to correct the error is to turn the face three more times. Both methods will return the face to exactly the same state it was in prior to the incorrect face turn.

Similarly, if you accidentally apply HedgeSlammer too many times, or start a HedgeSlammer in the wrong place, there are two ways to correct the error. One way to correct the error is to reverse every unintented turn that was made. The HedgeSlammer is a simple move-sequence, so reversing it is not too difficult. The other way to correct the error is to continue applying HedgeSlammers until a total of six HedgeSlammers have been applied. Applying six HedgeSlammers in a row will return the cube to its starting state. In other words, applying six HedegSlammers accomplishes exactly the same thing as doing nothing.

A HedgeSlammer can be applied to any two faces that intersect, regardless of the orientation of the cube. To perform a HedgeSlammer, first choose any two faces that intersect, then turn the first face clockwise, then turn the second face counter-clockwise, then un-turn the first face (i.e. turn it counter-clockwise), then un-turn the second face (i.e. turn it clockwise). Prior to applying a HedgeSlammer, it is usually easier to reorient the cube so that the first face that is turned is the Front face and the second face that is turned is the Right face.

It is technically possible to solve the cube using nothing but HedgeSlammers along with a total of three additional quarter-turns of a face. The three quarter-turns are needed to position the corner-cubies correctly. Once the eight corner-cubies are in their home positions, the remainder of the solve can be accomplished using nothing other than HedgeSlammers; no additional face turns are required. The method for doing so is not covered here.

The HedgeSlammer move-sequence (F R' F' R) has a mirror move-sequence (R' F R F') named SledgeHammer. The SledgeHammer could also be used to solve the cube, but minor adjustments would need to be made. For example, if SledgeHammer were used, edge-cubies would cycle in the opposite direction, and corner-cubies would twist in the opposite direction. This guide will only use HedgeSlammer and will never use SledgeHammer.

Position corner-cubies

The cube has eight corner-cubies - four in the top layer and four in the bottom layer - that must be moved to their home positions. The orientation of the corner-cubies when they are in their home positions is not important.

Choose any face of the cube and orient the cube to make that face the top layer. During the process of moving corner-cubies to their home locations, the top layer will never change. However, in later stages of the solve, the top layer will frequently change.

Move three three corner-cubies to their home positions in the top layer using any method you desire. (If the top layer already contains the four corner-cubies in their home locations, then move any one of the corner-cubies into the bottom layer. After doing so, ensure the other three corner-cubies in the top layer are in their home positions.)

Rotate the bottom layer so that the fourth corner-cubie is directly below its home position in the top layer. Apply a HedgeSlammer to move the fourth corner-cubie from the bottom layer into its home position in the top layer. This will also swap two diagonal corner-cubies in the top layer.

Applying a HedgeSlammer to move the fourth corner-cubie to its home position is convenient, but not required. Any method can be used to move the four corner-cubies into the top layer as long as the end result is that two diagonal corner-cubies are in their home positions and two diagonal corner-cubies are not in their home positions.

Since the four corner-cubies that belong in the top layer have been moved to the top layer, it necessarily means that the four remaining corner-cubies all belong in the bottom layer.

Turn the bottom layer any number of turns, in any direction until no corner-cubie is in its home position in the bottom layer.

There are four corner-cubies in the bottom layer that are not in their home positions and they are now ready to be moved to their home positions. Each corner-cubie in the bottom layer will be moved to its home position by moving the corner-cubie up into the top layer, rotating the top layer, and then moving the corner-cubie back down into the bottom layer. Use the following procedure to move the four corner-cubies in the bottom layer to their home positions:

1. Recall that HedgeSlammer swaps two corner-cubies along a vertical edge.
2. Use HedgeSlammer to move a corner-cubie from the bottom layer into the top layer.
3. Rotate the top layer until the corner-cubie is directly above its home position in the bottom layer.
4. Use HedgeSlammer to move the corner-cubie from the top layer into its home position in the bottom layer.
5. The previous step will have moved a new corner-cubie into the top layer.
6. If all corner-cubies in the bottom layer are in their home positions, then STOP, otherwise go to step 3.

Once all corner-cubies in the bottom layer have been moved to their home positions, examine the four corner-cubies in the top layer. If the four corner-cubies in the top layer are not in their home positions, rotate the top layer so that they are.

For clarity, the example below shows the corner-cubies in the bottom layer oriented and in the correct order (relative to each other), but none of the corner-cubies in the bottom layer need to be oriented or in the correct order (relative to each other).

When moving corner-cubies in the bottom layer to their home positions, do not follow the example above verbatim. The example above only works when all corner-cubies in the bottom layer are in order, relative to each other. When a bottom corner-cubie is in the top layer, the top layer must be turned so that the corner-cubie is directly above its home position in the bottom layer. This will require one or two quarter-turns in either direction.

The cube should now have all eight corner-cubies in their home positions.

The task of positioning the eight corner-cubies is now complete. One or more (usually many) of the corner-cubies will not be oriented correctly. Orienting the corner-cubies will occur in a later stage of the solve.

The process described above for moving the eight corner-cubies into their home positions will always work. If desired, one of two optional variations can be used. The advantage of using a variation is that 83% of the time (5 of 6 cases), it will be simpler and faster. The disadvantage is that it will not work 17% of the time (1 of 6 cases).

1. Configure the four corner-cubies in the top layer in the usual manner (two diagonal corner-cubies in their home positions and two diagonal corner-cubies not in their home positions). Once completed, instead of turning the bottom layer so that no corner-cubies are in their home position, turn the bottom layer so that exactly two corner-cubies are in their home position. This will not be possible in the case where all four corner-cubies in the bottom layer are in order, relative to each other.
2. Instead of configuring the four corner-cubies in the top layer in the usual manner (two diagonal corner-cubies in their home positions and two diagonal corner-cubies not in their home positions) configure the four corner-cubies in the top layer so that they are all in their home positions. If the bottom layer can be turned so that all four corner-cubies are in their home positions, then nothing more needs to be done. Otherwise, turn the bottom layer so that exactly one corner-cubie is in its home position (the other three corner-cubies will necessarily not be in their home positions). This will not be possible in the case where the bottom layer has two diagonally opposite corner-cubies that are in their home positions and two diagonally opposite corner-cubies that are not in their home positions.

In either case above, move the corner-cubies in the bottom layer to their home positions in the usual manner. (i.e. Move the corner-cubie into the top layer using HedgeSlammer, then rotate the top layer until it is directly above its home position, then use HedgeSlammer to move it into the bottom layer.)

Position edge-cubies

Once all eight corner-cubies have been moved to their home positions, all twelve edge-cubies must be moved to their home positions.

Applying HedgeSlammer swaps two horizontal corner-cubies in the top layer and also swaps two vertical corner-cubies. From this, we can deduce that if HedgeSlammer is applied two times in a row, both pairs of corner-cubies will swap places a second time and be returned to the same position they were at before the first HedgeSlammer was applied.

It does not matter how the cube is oriented, applying HedgeSlammer two times in a row always returns all corner-cubies to the same positions they started from.

From this point forward, always apply HedgeSlammer two times in a row. This will ensure that the corner-cubies do not change their positions. After applying HedgeSlammer two times in a row, the ending orientation of the corner-cubies will differ from their starting orientation, but the orientation of the corner-cubies should be ignored.

For the sake of brevity, applying HedgeSlammer two times in a row will be denoted as 'Hs2'.

If we examine how applying HedgeSlammer affects edge-cubies, we see that it moves three edge-cubies one step counter-clockwise in a three-step cycle. From this, we can deduce that if HedgeSlammer is applied three times in a row, the three edge-cubies will be returned to their starting position.

Since we must always apply HedgeSlammer two times in a row it means that each time we apply Hs2, the edge-cubies will be moved two steps 'forward' in a counter-clockwise direction. Since the edge-cubies move in a three-step cycle, this is the same as moving one step 'backwards'. In other words, applying Hs2 causes the edge-cubies to move one step clockwise in a three-step cycle.

By re-orienting the cube as needed and then applying Hs2, all edge-cubies can be moved to their home positions.

When moving edge-cubies, focus on one edge-cubie at a time and move it to its home position one step at a time by re-orienting the cube and then applying Hs2. If an edge-cubie is a long way from its home position, re-orienting the cube and then applying Hs2 must be done several times.

When moving edge-cubies to their home positions, do not think of the cube as having a fixed orientation. The cube should be freely re-oriented in whatever way is needed in order to move edge-cubies to where they need to go. (Do not re-orient the cube while applying Hs2, only re-orient the cube after applying Hs2. e.g. Apply Hs2, then re-orient, then apply Hs2, then re-orient, then apply Hs2, etc.)

Do not be concerned about the orientation of the edge-cubies. The edge-cubies only need to be moved to their home positions. Orienting the edge-cubies at this stage of the solve is optional, but can be done, if desired. Orienting the edge-cubies will be done in the next stage of the solve.

Each layer of the cube has four edge-cubies. When three edge-cubies have been moved to their home positions, you will find that it is impossible to move the fourth edge-cubie to its home position without dislogding one of the three edge-cubies. There are two methods for solving this issue: Pre-staging and Setup-Teardown.

Pre-staging and Setup-Teardown solve the same problem in different ways. The problem and solution is illustrated below.

Pre-staging is slow, but there is no danger of destroying past progress due to forgetting to do Teardown or doing Teardown incorrectly. Setup-Teardown can move edge-cubies to desired locations faster, but will undo past progress if done improperly.

Below are two more examples of how Setup-Teardown (a.k.a. Conjugate) can be used.

There are no restrictions on the moves for the Setup. The Setup can turn any number of faces any number of turns, however, it is critical that the Teardown undo all of the Setup moves in the reverse order. In practice, turning one face a quarter-turn is typically all that is needed for Setup.

There is a lot of freedom in how the edge-cubies can be moved to their home positions. When moving edge-cubies to their home locations, there is no correct order to place the edge-cubies.

Orient edge-cubies

At this point, all corner-cubies are in their home positions and all edge-cubies are in their home positions. The cube is still scrambled, but it is closer to being solved than it might, at first, appear.

When moving the edge-cubies to their home positions, applying Hs2 caused three edge-cubies to move one step clockwise in a three-step cycle. From that fact, we can deduce that if Hs2 is applied three times, the three edge-cubies will return to their starting positions. It is not so easy to deduce the ending orientations of the edge-cubies, but their ending orientations can be discovered by applying Hs2 three times in a row and observing how the edge-cubies are affected.

If Hs2 is applied three times in a row we find that the ending orientations of the three edge-cubies exactly matches their starting orientations. Since the edge-cubies start and end in the same positions, and since their start and end orientations are the same, we can see that applying Hs2 three times in a row accomplishes exactly the same thing as doing nothing. (This should not be too much of a surprise since applying Hs2 three times is exactly the same as applying HedgeSlammer six times.)

That discovery is not useful in orienting the edge-cubies, however, if we add a pivot to the mix, something useful happens.

Pivoting the cube clockwise is defined as re-orienting the cube so that the Up face becomes the Right face and the Right face becomes the Front face.

Pivoting the cube counter-clockwise is defined as re-orienting the cube so that the Up face becomes the Front face and the Front face becomes the Right face.

Pivoting clockwise twice is the same as pivoting counter-clockwise once. Pivoting counter-clockwise twice is the same as pivoting clockwise once.

Pivoting the cube is simply rotating the cube 120 degrees about an axis. The axis is defined by two corner-cubies. The first corner-cubie is located at the intersection of the Front, Right, and Up layers. The second corner-cubie is located at the intersection of the Back, Left, and Down layers (which is physically farthest away from the first corner-cubie).

No matter which direction the cube is pivoted (clockwise or counter-clockwise), applying Hs2 always and only affects the same seven cubies. And no mater how the cube is pivoted, each time Hs2 is applied it will always cause the same three edge-cubies to cycle one step clockwise. (Note that none of the animations below show the cube being pivoted. Instead different faces are rotated as if the cube were pivoted.)

Below is a list of pivot sequences. The list illustrates every possible way that pivoting can be combined with three applications of Hs2. The reason that Hs2 is applied three times is because it ensures that the three edge-cubies are returned to their starting locations. The list is for illustrative purposes only and can be ignored; there is no need to memorize this list.

As the nine illustrations above show, no matter which way the cube is pivoted, or how often the cube is pivoted, only two outcomes are possible: either two edge-cubies will be flipped (on the Up, Front, or Right face) or no changes will occur.

By flipping two edge-cubies at a time, all edge-cubies can be correctly oriented.

Due to the physical construction of the cube, there is no sequence of turns that can ever be devised which will flip exactly one edge-cubie. Edge-cubies can only be flipped in pairs. This means that there will always be an even number of edge-cubies that must be flipped in order to have all edge-cubies be oriented correctly.

Of the nine pivot sequences above, eight contain pivots and one contains no pivots. As previously discovered, if no pivots are performed then no edge-cubies are flipped. Since our goal is to orient edge-cubies by flipping them, we can discard the sequence containing no pivots. The remaining eight pivot sequences each contain one or more pivots. Of those eight, seven of them flip two edge-cubies and one of them flips no edge-cubies.

What this means is that, as long as you make sure to pivot at least once (and apply Hs2 three times), there is a 7/8 = 87.5% chance of flipping two edge-cubies. If you happen to apply the pivot sequence that flips no edge-cubies, then the cube will be in the same state that it started.

Unfortunately, there is no 'intuitive' way to know which pivot sequence will flip which edge-cubies. Here are several options for how to flip two edge-cubies:

• Use knowledge about how Hs2 affects edge-cubies. (See the boxed example below.) Each time Hs2 is applied, three edge-cubies move one step clockwise. One edge-cubie starts and ends on the Front face; one edge-cubie starts and ends on the Up face; one edge-cubie starts and ends on the Right face. The edge-cubies on the Front and Right faces 'rotate' about the center-cubie, while the edge-cubie in the Up face 'flips'. After applying Hs2 twice, examine the edge-cubies. The next (third) application of Hs2 will return all three edge-cubies to their home positions. Prior to applying Hs2 for the third time, pivot the cube clockwise or counter-clockwise. Pivoting changes the Front, Right, and Up faces, and thus controls which two edge-cubies are 'rotated' into their home positions and which edge-cubie is 'flipped' into its home position.
• While solving a cube, try a random pivot sequence and note which two edge-cubies flip.
  • If you happen to apply the pivot sequence: (Hs2 PivotCounterClockwise Hs2 PivotCounterClockwise Hs2 PivotCounterClockwise) then no edge-cubies will be flipped. This does not move the cube closer to being solved, but neither does it undo any progress that has already been made.
  • If you happen to apply any of the other seven pivot sequences then two edge-cubies will be flipped on the Front, Up, or Right face. You now know a pivot sequence that will flip two edge-cubies as well as the face where the flipping occurs.
• Memorize one of the seven pivot sequences above which flips two edge-cubies. Having to memorize an arbitrary pivot sequence goes against the spirit of this solution, but will work to orient all the edge-cubies.
• Apply a random pivot sequence. If it does not flip the two desired edge-cubies, then repeat the pivot sequence. Repeating a pivot sequence will undo the previous pivot sequence. After the cube has been restored, apply a different random pivot sequence. If the two desired edge-cubies are still not flipped, repeat the previous pivot sequence. Repeating the pivot sequence will undo the previous pivot sequence. Continue in this manner until the two desired edge-cubies are flipped. This technique is not recommended, but will eventually work.

Once you know a pivot sequence that flips two edge-cubies and the face where the flipping occurs, the cube can be re-oriented so that any two edge-cubies can be flipped.

When orienting edge-cubies, you do not have to worry too much about losing any progress that has been made. The worst that can happen when applying a pivot sequence is that two unwanted edge-cubies will flip. If you did not want those two particular edge-cubies to be flipped, simply repeat the previous pivot sequence. Repeating a pivot sequence will 'un-flip' whatever edge-cubies the previous pivot sequence flipped.

There is no correct order for flipping edge-cubies, but some orders require less work than other orders.

Setup-Teardown can be used to pair edge-cubies so that they can be flipped together.

Orient corner-cubies

When orienting edge-cubies we found that applying Hs2 three times in a row returns all edge-cubies to their starting position and orientation. Applying Hs2 three times in a row also returns all corner-cubies to their starting position and orientation.

We also know (or can observe) that each time Hs2 is applied, the corner-cubie in the bottom layer is twisted clockwise. And since Hs2 is applied three times, the corner-cubie gets twisted clockwise three times. This results in returning the corner-cubie to its original orientation. Instead of applying the three clockwise twists to a single corner-cubie, the three twists can be applied to two corner-cubies. This is accomplished by replacing the original corner-cubie with another corner-cubie.

In the example above, one clockwise twist was applied to the target corner-cubie, and two clockwise twists were applied to the replacement corner-cubie. We can use this technique to orient all corner-cubies.

The following procedure can be used to orient all corner-cubies:

1. Choose a corner-cubie that is already oriented. If needed, use the technique above to orient a corner-cubie.
2. Find the corner-cubie that is physically farthest away from the chosen oriented corner-cubie; this is the Replacement corner-cubie.
3. The Replacement corner-cubie will be used for every corner-cubie that needs to be oriented.
4. For each un-oriented corner-cubie (except the Replacment corner-cubie):
   1. Ensure the un-oriented corner-cubie is positioned bottom, right on the Front face.
   2. Ensure the Replacement corner-cubie is in the bottom layer.
   3. Apply the technique above to twist the un-oriented corner-cubie clockwise.
   4. If the corner-cubie is still not oriented correctly, apply the technique a second time.

The Replacement corner-cubie may start out oriented or un-oriented, and while orienting other corner-cubies, the Replacement corner-cubie may switch between being oriented and being un-oriented.

Orienting corner-cubies in the manner described above is effective, but slow. The process can be sped up if we are smarter about which corner-cubie is used as a replacement and when the replacement is used.

• Instead of designating a single corner-cubie as the Replacment, any corner-cubie in the bottom layer can be chosen as the replacement.
• Instead of twisting an un-oriented corner-cubie once and then twisting the replacement corner-cubie twice, the un-oriented corner-cubie can be twisted twice and the replacement twisted once.
• Instead of using one replacement corner-cubie, two replacement corner-cubies can be used. By doing so, it is sometimes possible to orient three corner-cubies at a time.

In some cases it will be impossible to position the cube so that two un-oriented corner-cubies are in the bottom layer. When that occurs, proceed to orient one of the un-oriented corner-cubies, in the usual manner, but choose any (oriented) corner-cubie in the bottom layer as the replacement corner-cubie. This will orient the un-oriented corner-cubie and un-orient the (oriented) replacement corner-cubie. The two remaining un-oriented corner-cubies can then be oriented in the usual manner.

An alternative method for handling this situation is to use Setup-Teardown.

In general, the steps for orienting a corner-cubie are:

1. Apply Hs2 until the corner-cubie in the bottom layer is oriented (this will always require applying Hs2 either once or twice).
2. Then replace the oriented corner-cubie (by turning the bottom layer),
3. Then apply Hs2 until the top and middle layers are restored (the replacement corner-cubie may still be un-oriented).
4. Then un-turn the bottom layer (which returns all corner-cubies to their home positions).

The process described above for orienting corner-cubies is simple, but somewhat laborious. A common technique that other solutions use to orient corner-cubies in the top layer is (R' D' R D). You may prefer using it over using Hs2 to orient corner-cubies.

Summary

A HedgeSlammer...

1. ...swaps two vertical corner-cubies between top and bottom layers.
2. ...swaps two diagonal corner-cubies in the top layer.
3. ...cycles three edge-cubies.
4. ...applied twice, twists a corner-cubie in the bottom layer.
5. ...applied six times is the same as doing nothing.
6. ...applied six times with a pivot, flips two edge-cubies.

Steps for solving the cube:

1. Position corners
   • Position three corner-cubies in top layer.
   • Use Hs to move last corner-cubie into top layer.
   • Turn bottom layer so no corner-cubies are in home positions.
   • Use Hs and the Up face to move corner-cubies in bottom layer to their home positions.
2. Position edges
   • Apply Hs2 and Pre-stage and/or Setup-Teardown.
3. Orient Edges
   • Apply (Hs2)3 and Pivots to flip two edge-cubies at a time.
4. Orient Corners
   • Apply (Hs2)3 and turn the Down face to orient corner-cubies in bottom layer.

Once you have mastered this method of solving the cube, you may wish to explore some variations.

Happy solving!