One Little Bat

One little bat goed off to play

And he saw two little pumpkins

And he said, "Hey, new friends!"

And then he saw some candy.

The Rubik's Cube (1/5)

PREFACE

There are two prerequisites for this course:

    An interest in the reasoning behind solving a Rubik's cube

    A viewing of the mathematical documentary, "The Year Without A Santa Claus"

        (For those of you who prefer reading, here's a synopsis.)

I maintain that the ability to solve a Rubik's cube is not a prerequisite, but it doesn't hurt.

The first step in solving a Rubik's cube is to solve just one piece.  Next, the solved area is expanded.  When the solved area gets big enough that it becomes difficult to solve anything more without destroying what's already solved (this is most commonly thought of as two out of three layers being solved), it becomes necessary to move just a few pieces at a time.  There are math tools that can help us, and that's what this post is about.

CHAPTER 1: INTRO

I got interested in the Rubik's Cube in 2016.
My goal at the time was to learn to solve the cube in a way that I would be able to remember many years later, which meant very few or no algorithms.  It worked; now it's 2022.

So that was the end of that, but recently my friend wanted me to make a Rubik's cube birthday cake for her son.  I did, and it was a disaster.  I accidentally re-baked my already beautifully flooded Rubik tile sugar cookies when I was trying to toast some pecans, and the rest was history.  To be fair, I knew I was set up for failure from the beginning; using black buttercream and constructing a cube out of cake doesn't set one up for success.  And while all of this was happening, my sister asked me to write a post like this.  Just like this.  I'm sure she will be very pleased.  (K, maybe not just like this.)

I'd like to take a moment to emphasize that this post is not written for speedcubers.  I have absolutely no interest whatsoever in speedcubing.  I'm sure it requires a lot of intelligence and practice and memorization and dexterity, but... why?  There are people who do this, so why would anyone want to compete with them?  I'd only memorize over 20 algorithms (speedcubers regularly memorize hundreds) if I were the only person in the world doing it, and then maybe I could go on the Ellen Show and get rich and then start teaching a college course on what it takes to memorize over 20 algorithms.  Is that how money works?  Then I could see it being worth it.

I'd also like to mention that anything I write will pretty much be notes on Ryan Heise's website because that's the source I found the first time, and that's the source I found this time too.

Here it is in all its glory: https://www.ryanheise.com/cube/

So concise, so well explained, so organized, and with such a helpful teaching graphic at every turn.  I have a huge crush on that website, and if I had long lush eyelashes, I would bat them suggestively towards that tab right now.

The 3x3x3 Rubik's cube is a puzzle that can be solved all sorts of ways.  The most popular are:

    Petrus by Lars Petrus (1980's)

    CFOP by Jessica Fridrich (1997)

    ZB by Zbigniew Zbrorowski and Ron van Bruchem (2002)

    HTA/Human Thistlethwaite Algorithm by Ryan Heise (2002)

    Heise by Ryan Heise (2003)

    Roux by Gilles Roux (2003)

    ZZ by Zbigniew Zborowski (2006)

If you want to know more about those methods, here's a great source for that.

In 2016, I learned the corner 3-cycle from Heise's website, and found that that's all I needed for the Petrus, so that's mostly all I do from time to time, and very slowly.  But back to what we all came here for: me plagiarizing Heise.

CHAPTER 2: NOTATION

The little pieces of a Rubik's cube are called cubies.

Each square of 9 cubies is called a face.

The cube has a 6-part center piece, 12 edge cubies, and 8 corner cubies.

Center cubies each have 1 color tile, edge cubies each have 2, and corner cubies each have 3.

All together, that's 54 color tiles, which = 9 tiles x 6 faces.

White is opposite yellow, red is opposite orange, and blue is opposite green.

This is called the "minus yellow" color scheme:

    Yellow - yellow = white; Orange - yellow = red; Green - yellow = blue.

When white is up and Red is on the Right, Blue is on the Back.

    That's just how they made it.

The faces are called up, down, right, left, front, and back, notated: U D R L F B.

The rotation U means a 90° clockwise turn of the U face when looking (down) at it.

The rotation D means a 90° clockwise turn of the D face when looking (up) at it.

Take a moment to review what clockwise means for D and B in particular.

Notice that for opposite planes, "clockwise" means opposite directions.

U² is a 180° turn, and U^-1 is a 90° counterclockwise turn.

And so it is with all the other faces.

CHAPTER 3: GROUP THEORY

And now, I'm going to define the term "group" in the world of math.

A group consists of elements and a binary operation that can be applied to the elements.

Let's pretend our group has elements X, Y, and Z, and our operation looks like this: *.

The four properties of a group are:

    1.  The group is closed, meaning X*Y = G, and G is in the group.

    2.  The operation is associative, meaning (X*Y)*Z = X*(Y*Z).

    3.  The group contains an identity element (I) such that X*I = I*X = X.

    4.  Each element has an inverse (X^-1), meaning X*X^-1 = X^-1*X =  I.

If the operation is always commutative (X*Y = Y*X), then the group is called abelian.

This is extremely relevant to the Rubik's cube.

The elements of Group Rubik are the different sequences of rotations.

The operation of Group Rubik, *, means "and then," so U*F means "rotate U and then rotate F."

The identity element in Group Rubik is to perform no operation.

For example, "R" is an element of Group Rubik, as is "B^-1*U*B*D^-1*B^-1*U*B*D."

For each configuration of the Rubik's cube, there's a series of instructions that can get you there.

That means for every configuration of the Rubik's cube, there's an element in Group Rubik.

Group Rubik is non-abelian; U*F is not the same as F*U (wink).

Sometimes our operation is commutative; R*L = L*R.

That's because there's no cubie intersection between rotation instructions R and L.

Restricting the operations of Group Rubik in some way generates a smaller group that still has all the properties of a group, and that's called a subgroup.  The different Rubik subgroups maintain different things; some preserve edge orientation, some preserve solved blocks, etc.

Any sequence of rotations on the Rubik's cube, if repeated a particular number of times, creates a cyclical path for each piece it moves.  This means that any sequence of rotations, when repeated enough times, restores the entire cube to its initial position!

Let's take a look at the simplest case, when a sequence is a single 90° rotation like "F."  All four edge cubies and all four corner cubies travel on plane F, and the sequence must be repeated four times for each cubie to be restored to its original position.  Because edge cubies never move to corner slots and corner cubies never move to edge slots, I like to think of this as two simultaneous cycles, each of length four.

The number of times a sequence of rotations must be repeated to restore the entire cube to its initial position is called the order of that sequence, and the order is found by calculating the least common multiple of the lengths of all the cycles generated by the sequence.  In the case above, the least common multiple of (4, 4) is 4, so that's the order of that sequence.

Usually, not one cycle but several are generated by a sequence.  Let's take a look at F²*U².  Performing the sequence one time disrupts all seven edge cubies and six corner cubies involved.  The second time solves four of the edge cubies and no other cubies; the third un-solves those four edge cubies, solves the other three edge cubies, and also solves all six corners.  So in this case, there is a cycle for four edge cubies (length 2), a cycle for three edge cubies (length 3), and a cycle for six corner cubies (also length 3).  The order for sequence F²*U² is the least common multiple of (2, 3, 3), which is 6.

The sequence F*U² generates a five edge cubie cycle (length 5), a two edge cubie cycle (length 6), and a six corner cubie cycle (also length 6).  The order for sequence F*U² is 30.

The sequence F*U generates a seven edge cubie cycle (length 7) and a six corner cubie cycle (length 15), and has an order of 105.

If you already know how to solve a cube, you could try any of these subgroup exercises.

Otherwise, give it a shot with my favorite simulator.

    (F*U) 105 times

    (F*U^-1) 63 times

    (F²*U²) 6 times

    (F*U*F^-1*U^-1) 6 times

    (F*U*F*U^-1) 5 times

Michael Gottlieb's order chart can give you more sequences to explore.

Also, his order calculator can generate the order of any sequence you'd like to try.

Note that the highest order on the 3x3x3 is 1,260.

CHAPTER 4: TERMS & THE LAWS OF THE CUBE

An orientation refers to what direction the cubie colors face.

    orient - to change the direction a cubie's colors face

A permutation refers to where the cubies are positioned.

    permute - to move in a permutation

A conjugate is a sequence of the form N*M*N^-1.

A commutator is a sequence of the form X*Y*X^-1*Y^-1.

Notice that N, M, X, and Y can represent any element of Group Rubik.

The job of a conjugate is to:

    1.  Bring something somewhere

    2.  Do something to it

    3.  Put the new altered thing back

The job of a commutator is to:

    1.  Do nothing at all if the group is abelian

    2.  Otherwise, shuffle around the things in the intersection of X and Y

    

A permutation can be the action of changing an arrangement, or the arrangement itself.

For now, think of a permutation as a position, solvable or not, of the cubies on the Rubik's cube.

If two elements of Group Rubik result in the same oriented permutation, they are equivalent.

There are about 43 quintillion solvable oriented permutations, and a quintillion is 10¹⁸.

That number is so big that it's pure nonsense.

In fact, it's such nonsense that while a quintillion = 10¹⁸ here, it's 10³⁰ in the UK.

Sure, why not, that's fine.

Let's consider for a moment only the placement of cubies, and not their orientations.  Assuming that edge cubies are only placed in edge slots and corner cubies are only placed in corner slots, only half of the cube positions you can imagine would allow you to reposition each cubie back into its original slot.  Remember, because we're ignoring orientation right now, we're not necessarily imagining a true return to the original position.

A swapping of elements of a pair of like cubies is called a transposition, so each 90° rotation of a face creates 3 edge cubie transpositions and 3 corner cubie transpositions.  (For further reading on this, please see chapter 3 of my sequel post.)  That means that for any number of rotations, the number of edge cubie transpositions and the number of corner cubie transpositions are either both odd or both even.  If you were to add up the total number of edge cubie and corner cubie transpositions, you would always get an even number of total transpositions.

But you could imagine a configuration with an odd number of total transpositions.  For example, imagine picking two edge cubies out of a solved cube, exchanging them, and popping them back in.  That would be an odd number of transpositions in the edge cubies (1) with an even number of transpositions in the corner cubies (0), so that's no longer a solvable position Rubik's cube.

Only half the edge orientations you can imagine are solvable, because each rotation flips an even number of edges.  But again, you could imagine an odd number of flips, like an otherwise solved cube with one edge cubie reoriented in its correct position.

Only a third of the corner orientations you can imagine are solvable.  If you imagine an otherwise solved cube, there are two wrong ways to reorient any corner cubie.

If only half of the placements of cubies you can imagine could be solvable due to the requirement of an even number of transpositions, and only half of those are solvable due to edge orientation constraints, and only a third of those are solvable due to corner orientation constraints, then altogether, if you deconstructed a correctly solved cube and reconstructed it at random, the resulting permutation would only be solvable (1/2)x(1/2)x(1/3) = 1/12 of the time.

To find all solvable and unsolvable permutations, 12 times 43 quintillion should be:

(3x8)x(3x7)x(3x6)x(3x5)x(3x4)x(3x3)x(3x2)x(3x1)

    which are all the places we could imagine putting the corners,

TIMES

(2x12)x(2x11)x(2x10)x(2x9)x(2x8)x(2x7)x(2x6)x(2x5)x(2x4)x(2x3)x(2x2)x(2x1)

    which are all the places we could imagine putting the edges,

TIMES

(1)

which is every place we could imagine putting the center.

(3⁸x8!)x(2¹²x12!)x(1) = 5.190240392938783 x 10²⁰,

And when we divide that by 12, we get 4.325200327448986 x 10¹⁹.

Mic drop.

CHAPTER 5: PLAY TIME

Grab a solved cube!

Before we begin, it's important to choose your very favorite cube orientation.

First, put the most memorable color on the bottom.

From there, select your favorite corner and remember its position.

Make sure you return to this cube orientation before performing any sequence of rotations.

Once you've graduated from this course, you will have earned your freedom.

My friend Jack says, "This thing [the Rubik's cube] is dynamite!"

It's not a compliment; it means BE CAREFUL.

Play time can be dangerous.

(If your cube explodes, just text me, and we'll get you back up and running in a jiffy.)

Level 1

Let's try our first conjugate - I'll go back to F*U since it's on the mind (so cheeky).

F*U*F^-1

Great.  Let's undo that.

Can you figure out what (F*U*F^-1)^-1 would be?  Of course you can!

To find an inverse, you start at the end, move backwards, and reverse directions.

F*U^-1*F^-1 gets us back to where we started.

Some of you may already know the logic terms inverse, converse, and contrapositive.

Here they are for review.

    Statement: if you're still reading this post, then you're a nerd.

    Inverse: if you're not still reading this post, then you're not a nerd.

    Converse: If you're a nerd, then you're still reading this post.

    Contrapositive: If you're not a nerd, then you're not still reading this post.

Notice that the inverse and converse of true statements aren't necessarily true.

However, the contrapositive of a true statement is always true.

Constructing an inverse in the world of cubing is a lot like the contrapositive:

You must both reverse directions and also negate!

So, nerds, here's some homework.

Please review the terms "conjugate" and "inverse" as it applies to cubing.

Take out a pen and a pad of paper, and work through the following:

    1. What would the inverse of (F*U*F^-1)*D*(F*U^-1*F^-1)*D^-1 be?

        Answer: D*(F*U*F^-1)*D^-1*(F*U^-1*F^-1)

    2. What would the inverse of (R^-1*D*R*F*D*F^-1)*(U)*(F*D^-1*F^-1*R^-1*D^-1*R)*(U^-1) be?

        Answer: (U)*(R^-1*D*R*F*D*F^-1)*(U^-1)*(F*D^-1*F^-1*R^-1*D^-1*R)

    3. If N = L, X = F, and Y = U

        First construct a commutator of the form X*Y*X^-1*Y^-1.

        Then use that commutator as the center (M) of a conjugate of the form N*M*N^-1.

        Write down this sequence of rotations as well as its inverse.

        Answer: L*(F*U*F^-1*U^-1)*L^-1;  Inverse: L*(U*F*U^-1*F^-1)*L^-1

If all your answers were correct,

Level 1 conjugate and inverse achievement unlocked.

Level 2

Let's try our first commutator.

F*U*F^-1*U^-1

Hm, that messed up 7 cubies because that's how many cubies were caught in the mix.

Let's undo that by making another inverse.

(F*U*F^-1*U^-1)^-1 = U*F*U^-1*F^-1.

You're back to a solved cube, right?

Review: let's return to our first exercise.

Repeat F*U*F^-1 and look around your whole cube.

One face has only one wrong cubie, and one wrong cubie is a very good thing.

Can you find which face it is?  Of course you can - it's D.

We've noticed that F*U*F^-1 is an operation that changes one corner on the down side.

Meanwhile, it'd be nice to keep this "progress" but undo the other damage.

The only face we can change that won't get in the way of fixing those other 7 cubies is D.

In Level 3, we'll try making a commutator using F*U*F^-1 and D or D² or D^-1 together.

Who knows what could happen.

So restore your cube; F*U^-1*F^-1 worked before, and it shall work again.

Please review the term "commutator."

Please compare the terms "commutator" and "conjugate."

Level 2 commutator and review achievement unlocked.

Level 3 Intro

Treating F*U*F^-1 as X, and D as Y, our new commutator X*Y*X^-1*Y^-1 would be:

(F*U*F^-1)*D*(F*U^-1*F^-1)*D^-1

Try it.
Surprise!
You just did the corner 3-cycle that I learned in 2016, and that's very nearly everything I know.
An n-cycle is a permutation that moves n cubies around in a cycle.
So for a 3-cycle, cubie 1 goes to where cubie 2 was, 2 to 3, and 3 to 1.

To re-solve your cube, choose your own adventure:

    the inverse you can construct above, or

    read on to try the corner 3-cycle in chapter 10.

Level 3 to be continued...

CHAPTER 6: COMMUTATORS

How did that happen?

Let's say a sequence of rotations, X, affects a bunch of cubies.

A different sequence of rotations, Y, affects another bunch of cubies.

The cubies that were affected by both X and Y are called the intersection.

Let's return to the fact that Group Rubik is non-abelian.

For a moment, let's pretend it is abelian.

An abelian Group Rubik would mean that X*Y = Y*X.

In abelian Group Rubik, the commutator X*Y*X^-1*Y^-1 = X*X^-1*Y*Y^-1 = I*I = I.

X and Y would commute entirely, and there would be nothing in the intersection.

But in reality, Group Rubik is non-abelian.

A commutator for a non-abelian group measures what did not commute between X and Y.

We want to make what did not commute between X and Y as small as possible.

If only one cubie is permuted by both X and Y, the commutator X*Y*X^-1*Y^-1 cycles 3 cubies only.

Such a procedure is called a 3-cycle.

As Jamie Mulholland explains, notice the position that's the intersection of X and Y.

The commutator will affect:

    1. the cubie originally in that position

    2. the cubie that X brings into that position, and

    3. the cubie that Y brings into that position.

And that's it.

CHAPTER 7: CONJUGATES

Conjugates, on the other hand, are much easier to understand.  Simple conjugates consisting of three rotations like F*U*F^-1 are hardly talked about as conjugates at all, and are referred to as "intuitive" moves.  They play a large role in blockbuilding (see chapter 16).

Conjugates are of the form N*M*N^-1, and M usually represents some commutator.  While M is doing all the heavy lifting, N just gets the to-be-cycled cubies into position, and N^-1 fixes what N messed up.  I love conjugates, and although it makes perfect sense that they work, I still think they're kind of magical.  If someone gifted me a pet unicorn, now we all know what I'd name it.

CHAPTER 8: 3-CYCLE COMMUTATORS ON THE CUBE

To create a 3-cycle commutator on the cube, two of the three cubies we're cycling must be in the same plane of rotation.  Usually that's an outer plane, but it could be a "middle slice" as well (see chapter 12).  For now, we'll just assume it's an outer plane.  The third cubie will be somewhere else - somewhere we think of as "away from" or "opposite" the first plane of rotation.

We already know that commutators are of the form X*Y*X^-1*Y^-1.  Either X or Y must replace one of the two cubies on the two-cubie-holding plane of rotation without otherwise disturbing that plane.  When this happens, it replaces that cubie with the one from "somewhere else."  The rest of the cube will get messed up, and that's okay.

The other part of the commutator, X or Y, rotates the two-cubie-holding plane of rotation to shift the untouched cubie into the place where the "cubie from somewhere else" has just arrived.

I've presented these commutators so that X is a 3-move sequence and Y is a 1-move sequence.  In these cases, "X" is the part of the commutator that replaces one of the two cubies on the two-cubie-holding plane, and "Y" is the part that rotates that plane.

When we begin a 3-cycle, there are two cubies on a plane of rotation.  We'll call those two cubies Vixen and the Mayor of Southtown, and we'll call the plane of rotation Snowmiser's.  There's also another cubie from "somewhere else," which will be the opposite plane of rotation from Snowmiser's.  We'll call the third cubie Mrs. Claus, and the opposite plane of rotation Heatmiser's.  Notice that Snowmiser's and Heatmiser's are planes, while Vixen, the Mayor, and Mrs. Claus are cubies.

Our first cubie is Vixen, and she's always our leader.  If you've taken care of your prerequisites for this course, you will already know that Vixen gets ill when the temperature is too high.  She always wants to stay at Snowmiser's, so she always wants to move from her original position over to the Mayor's slot.  This immediately defines the direction of our 3-cycle.  If Vixen replaces the Mayor, the Mayor must replace Mrs. Claus, and Mrs. Claus must replace Vixen.  That's everything there is to know about the permutation.

Regarding orientations, Vixen must be ready to arrive at the Mayor's slot, correctly oriented after a Snowmiser rotation.  This is Vixen's defining characteristic - her orientation is "already correct" in regards to the Snowmiser plane, and it's by far the easiest to check.

Our second cubie is the Mayor.  The Mayor is always freezing in Snowmiser's, and he always wants to get out of there, and over to Heatmiser's.  We don't care about him too much because he's annoying.

Our third cubie is Mrs. Claus.  She's comfortable anywhere.  She starts off at Heatmiser's and ends up at Snowmiser's in Vixen's original slot.  We do check in on her orientation any time we need control over the orientations of all three cubies, such as the moment we face the final three unsolved cubies.

According to the laws of the cube, we only need to check 2 out of the 3 orientations, and the last one will take care of itself.  That means we could choose to check either Mrs. Claus' or the Mayor's, but Mrs. Claus' orientation is easier to deal with, and according to our principles, we don't care about the Mayor anyway.

Returning to the overview of the commutator, Vixen and the Mayor begin in Snowmiser's, while Mrs. Claus is in Heatmiser's.  Between X and Y, one of them will often be three moves, and the other one will often be one.  Assuming X has three moves, the X in our commutator will pull Vixen out of Snowmiser's and put Mrs. Claus in Vixen's original slot without otherwise disturbing Snowmiser's.  The rest of the cube will get messed up a bit.

The Y in our commutator will shift the Mayor to where Mrs. Claus just arrived.

The inverse of X will pull the Mayor out of Snowmiser's and put Vixen in the Mayor's original slot without otherwise disturbing Snowmiser's.  The rest of the cube will get solved a bit.

The inverse of Y will rotate Snowmiser's back to its original position.

CHAPTER 9: WARNING - NO LIFEGUARD ON DUTY

I'm not going to pretend that this is easy; in fact, I have been told as much by George and my dear ol' Dad.  If you're a bit worn out at the moment, now would be a good time to take a break.  Come back when you're well rested, or you've refreshed yourself with a prairie oyster.

If you're back, wow.  Welcome back!  Don't lose your footing, my strong-willed warrior.  Start by re-reading chapters 7 & 8 to prepare you for your journey.  They aren't there to be entertaining.

From here on out, please work slowly, using a pen and pad of paper to keep track of what you're doing.  Remember your favorite cube orientation and use it.  With good records, you can always take the inverse of what you did.  That being said, it's very easy to make mistakes.  If you do find yourself with a scrambled cube you can't solve, fear not.  You still have two options:

    1.  Scroll down to chapters 16 & 17.

        Practice block building until a knight in shining armor who solves Rubik's cubes appears.

        I'm sure you'll accidentally either run into one or become one yourself.

    2.  If you're tired of that because no knight is appearing (this happened to Princess Fiona),

        It's time to disassemble your cube.

        I promise it's not considered cheating unless you give up on this post.

        For the traditional Rubik's,

            Place the cube onto a flat surface and rotate the top face 45°.

            Holding the cube securely, wedge the tip of a butterknife under a top edge cubie.

            Push the handle down, popping the edge cubie up and out of the cube.

            Disassemble the entire cube.

            Reconstruct a solved cube, ending with a bit of force on the last top edge.

        For speed cubes,

            Find a center tile with a secret trap door on the side.

            This is probably the white center cubie; on the Roxenda, it's the tile with a logo.

            Use a small flathead or a fingernail to pry off the tile.

            Use a small phillipshead to unscrew the screw underneath.

            Disassemble the entire cube.

            Reconstruct a solved cube, screwing everything into place and popping the tile back in.

CHAPTER 10: THE CORNER 3-CYCLE

If you chose the inverse to solve your 3 corners, reset your Level 3 by repeating

(F*U*F^-1)*D*(F*U^-1*F^-1)*D^-1.

You should have three corner cubies out of place on an otherwise solved cube.  There are a few ways to understand the relationship these cubies have to each other and to the cube here; in this particular set up, they could all share a face, which is the left face.  But to perform a 3-cycle, we have to make sure we're seeing the cube as having two cubies on one face and the other on an opposite face.  We also have to make sure two of our three unsolved cubies are oriented to our liking.  If we're not very careful about our first two moves, our 3-cycle might not solve all three of our corners.

Before we begin to re-solve the cube, I'm going to talk you through this.

Make no further rotations until I signal that we're clear for takeoff!

As described above, the two opposite planes we're using will be called Snowmiser's and Heatmiser's.  We're going to rotate 3 corners around.  Corner 1 is Vixen, corner 2 is the Mayor of Southtown, and corner 3 is Mrs. Claus.

When we begin, Vixen and the Mayor are hanging out at Snowmiser's, and Mrs. Claus is having cocoa over at Heatmiser's.  Vixen wants to shift over to the Mayor's slot; the Mayor, who's freezing, is very anxious to catch a ride to Heatmiser's; and Mrs. Claus wants to take Vixen's place.

Vixen will travel to Heatmiser's.

Mrs. Claus will take her place.

Mrs. Claus will then travel back to Snowmiser's.

We can think of this as Vixen and Mrs. Claus changing places.

That 3 move exchange makes up the X in our commutator.

Then we'll rotate Snowmiser's.  That's the Y.

Then the Mayor and Vixen will change places in 3 moves.  That's X^-1.

Then we'll rotate Snowmiser's back to its original place.  That's Y^-1.

To elaborate on X, first we'll pull our correctly oriented Vixen out of Snowmiser's and over to Heatmiser's.  At Heatmiser's, we'll replace Vixen with Mrs. Claus by rotating Heatmiser's.  Finally, we'll bring Mrs. Claus back to where Vixen was by reversing our first rotation.

To elaborate on Y, we'll rotate Snowmiser's until the Mayor shifts to where Mrs. Claus landed.

For the rest of the commutator, we just need to remember how to construct inverses.

Level 3

Here are our clues; you're Sherlock.

    1.  Corner 1 Position & Orientation

        Vixen must share a plane with the Mayor and not Mrs. Claus.

        The color of one of her 3 tiles must be aligned with that face.

            That face will become Snowmiser's.

            So Vixen's Snowmiser tile must be on the Snowmiser plane!

            It looks like we have two choices for Vixen right now:

            Corner DLF matches the L face and corner DLB matches the D face.

    2.  Corner 2 Position

         The Mayor must be at Snowmiser's with Vixen.

            Either Vixen option shares a Snowmiser's with a Mayor.

    3.  Corner 3 Position

         Mrs. Claus, who belongs in Vixen's slot, must be at Heatmiser's.

    4.  Corner 3 Orientation

         After Vixen comes to Heatmiser's, Mrs. Claus must be oriented correctly to take her place.

            Remember, Mrs. Claus only moves with a Heatmiser rotation.

            After Vixen's move, we want Mrs. Claus' Snowmiser color out of Heatmiser's plane.

As Sherlock, you have deduced:

            If Vixen = DLF, Heatmiser's would be the R face, and there's no unsolved corner on R.

            If Vixen = DLB, Heatmiser's would be the U face, and there is an unsolved corner on U.

            That means Vixen is DLB and Mrs. Claus is ULF.

            That also means Snowmiser's is D, Heatmiser's is U, and the Mayor is DLF.

And if those conditions are met, we're clear for takeoff!

Move Vixen to Heatmiser's in such a way such that:

    Mrs. Claus remains in Heatmiser's, and

    Mrs. Claus is correctly oriented to replace Vixen.

Note that Vixen can always travel to Heatmiser's in 2 different ways.

    (In our case, we need to use B^-1, not L.)

Move Mrs. Claus to where Vixen was with a Heatmiser rotation.

Reverse rotation 1.  Those 3 rotations together are our X.

Rotate Snowmiser's so that the Mayor is where Mrs. Claus just landed.  That's our Y.

Reverse rotation 3, reverse rotation 2, and reverse rotation 1.

That exchanged Vixen with the Mayor and completed our X^-1.

Reverse rotation 4, restoring Snowmiser's.  And that's our Y^-1.

Altogether, our commutator was:

(B^-1*U*B)*D^-1*(B^-1*U^-1*B)*D

Level 3 achievement unlocked?

Compare it to the inverse just for fun:

D*(F*U*F^-1)*D^-1*(F*U^-1*F^-1)

In the inverse,

    Vixen goes to the Mayor's slot (D)

    Vixen and Mrs. Claus trade places (F*U*F^-1) 

    The Mayor travels to where Mrs. Claus landed (D^-1)

    Vixen and the Mayor trade places (F*U^-1*F^-1)

After you learn any commutator, you can always think in terms of its inverse as well.

Right about now you might be wondering, "But Lan, what if those conditions weren't met?"

Well, that's okay too, because we can always turn our commutator into a conjugate!!

Let's call this (3+1+3+1) 8-move sequence "M."

If we find a rotation or rotations "N" that move our 3 corners in such a way that they now meet the 3 position and 2 orientation conditions, then N*M*N^-1 will still solve those corners.  That's what chapter 7 was about.

To review the corner 3-cycle, here's the original source.

CHAPTER 11: THE 2 CORNER TWIST

Now that we've figured out how to permute some corners, let's figure out how to reorient one.

According to the laws of the cube, reorienting one will reorient another, so we'll be reorienting two.

Start with a solved cube.

This is an overview of the plan; don't rotate anything until we're in Level 4.

We'll need to reorient a corner cubie while altering nothing else on one of its three planes.

We'll call that plane the "preserved face."

Also, the second corner cubie that will be reoriented will share the preserved face.

    First we'll pull a corner from its slot into the opposite plane.

    Then we'll move it out of the way.

    Then we'll fix the preserved face by inverting our first rotation.

There are specifics to how this is done, and we'll get into the details shortly.

That will have been 3 moves.

Then we'll put that corner back into its slot but through different rotations so that it's reoriented.

That will have been another 3 moves, so all of that together will have been 6 moves.

The 6 moves together are our X.

Rotating the preserved face to select a second corner to reorient will be our Y.

Finishing the commutator should fix everything we just messed up.

Level 4

We'll try this first with an oriented example because I'm very generous.

I'm right handed, so I'll reorient the corner cubie in the up-right-front position, URF.

U will be the preserved face, also known as the rotation in Y.

Please follow along.

First, we'll bring URF down with R^-1.

Then we'll move it out of the way with D.

Then we'll fix the top with R.

Notice that R brought our corner cubie into an adjacent corner slot from its initial position.

That's because we had to choose D instead of D^-1.

Had we pulled URF down with F, we would have had to choose D^-1.

Next, we'll pop it back into place but through a different sequence.

That means we'll use F*D*F^-1.

Those 6 moves together were our X.

Now we need our Y; operations U, U² or U^-1 all work for this.

I'll go with U.

Notice that we're choosing which second corner to reorient with that rotation.

To finish our commutator, we have to invert our X:

(R^-1*D*R*F*D*F^-1)^-1 = F*D^-1*F^-1*R^-1*D^-1*R.

Then we need to finish with Y^-1, and that would be U^-1.

(R^-1*D*R*F*D*F^-1)*(U)*(F*D^-1*F^-1*R^-1*D^-1*R)*(U^-1)

How did you do?  With a pad of paper and nothing but the concept to guide you, can you make your own 2-corner twist?  Perhaps on the same corners you just twisted?  If you repeat the algorithm we just did exactly, you'll need to do it twice before those corners are correctly oriented again.  To restore the corners in one algorithm only, you could either try to be clever and rotate URF down with F first instead of R^-1, or simply take the inverse of what we just did.

Re-solve your cube.

Level 4 achievement unlocked?

Let's review the 2 corner twist once more, but in a more general way.  Snowmiser's and Heatmiser's are any two opposite planes, and when our story begins, the cube is already solved with the exception of two corner cubies named Jingle Bells and Jangle Bells.  They are both occupying Snowmiser's.  They're in their correct slots, but require reorienting.

We pulled corner Jingle Bells out of Snowmiser's and into Heatmiser's with one of two possible plane rotations.  We then moved him out of the way by rotating Heatmiser's in such a way that Jingle Bells stayed along the plane of our initial rotation.  Reversing our first rotation restored Snowmiser's edge cubies and placed Jingle into Heatmiser's, just across from his original slot.

Some new corner cubie that we don't care about took Jingle's original position.  We pulled that corner back down to Heatmiser's again, but with the other plane of rotation than the first one we first chose for Jingle.  We rotated Heatmiser's to replace the corner cubie we don't care about with Jingle.  Then we reversed our fourth rotation, which restored Snowmiser's with a reoriented Jingle.

All 6 of those moves were our X.

Next, we rotated Snowmiser's to select our Jangle, and that was our Y.

We inverted our X, which reoriented Jingle and also solved what we had messed up.

Finally, we inverted our Y to straighten up Snowmiser's.

Note that the initial plane rotation that we chose affects whether Jingle twists clockwise or counterclockwise.  According to the laws of the cube, Jangle will twist the other way to maintain the balance.  If it's difficult to figure out which plane of rotation to begin with, just pick one for now and commit to it through the commutator.  You will have reoriented both Jingle and Jangle once, and you can certainly do it again by repeating the exact same sequence of rotations if need be.

To play with the 2-corner twist, here's the original source.

CHAPTER 12: THE EDGE 3-CYCLES PART I

So far I've considered myself too lazy for "middle slice" moves, which is a totally reasonable way to be, but somehow I find myself writing chapter 12.  I guess it's time to start notating the middle slices, and using them to boot.

Middle slices are the planes that are in the middle layers of the cube, each containing 4 center cubies and 4 edge cubies.  Put another way, if you were to rotate two opposite planes so that they moved as one unit in the same direction, and then you reoriented your cube so the two opposite planes were back to where they had started, the plane in between those planes would have shifted.  That's a middle slice move.

I'll recognize 3: middle up (MU), middle right (MR), and middle front (MF).

MU will move like U but one layer underneath,

MR will move like R but one layer to the left, and

MF will move like F but one layer behind.

Now that we have middle slice moves, this chapter will cover three related edge 3-cycles:

    1.  Level 5, in which our middle slice is the first move in our 3-move X

    2.  Level 6, in which our middle slice is the second move in our 3-move X

    3.  Level 7, in which our middle slice is our 1-move Y

When Vixen is an edge cubie, sometimes it's easier to think of her as being oriented in such a way that she's ready to rotate on an outer plane (sometimes I will simply call these "planes") to get to where she needs to go, which is exactly what we did with the corner 3-cycles.  Other times, it's easier to think of her as being oriented in such a way that she's ready to rotate on a middle slice.

In Levels 5 and 6, Vixen is oriented to rotate on an outer plane, and in Level 7, she's oriented to rotate on an middle slice.

There's a set of four adjacent planes, each of which is neither Snowmiser's nor Heatmiser's.  Let's call that band of new lands Mothernature's.  The Polar Express runs through all three dimensions of the cube: the equator of Mothernature's, which never reaches Snowmiser's or Heatmiser's, and also the two train tracks connecting Snowmiser's and Heatmiser's.  Any middle slice movement will be considered a ride on the Polar Express.

Ready?

Level 5

When our first story begins, edge cubies Vixen and the Mayor are hanging out at Snowmiser's just as before.  And just as before, Vixen wants to take the Mayor's slot, the Mayor wants to get out of there because he's freezing, and Mrs. Claus, who's also an edge cubie, wants to take Vixen's place.  It should come as no surprise that we need to check the positions of all three edge cubies and the orientations of two.

    1.  Edge 1 Position & Orientation

        Vixen must share a plane with the Mayor; Mrs. Claus is not on this plane.

        The color of one of her 2 tiles must match the color of the rest of that face.

            That face will become Snowmiser's.

    2.  Edge 2 Position

         The miserable Mayor must be at Snowmiser's with Vixen.

    3.  Edge 3 Position 

         After Vixen takes the Polar Express to Heatmiser's, Mrs. Claus must be in Heatmiser's.

            Sometimes Mrs. Claus is across Vixen at Heatmiser's before Vixen travels.

            Sometimes she's waiting at Heatmiser's already.

    4.  Edge 3 Orientation

         After Vixen arrives in Heatmiser's, Mrs. Claus must be correctly oriented to take her place.

            Mrs. Claus can only take Vixen's place with a Heatmiser rotation.

From a solved cube, set yourself up with:

(R^-1*MU^-1*R)*D*(R^-1*MU*R)*D^-1

Then choose edges Vixen, Mrs. Claus, and the Mayor to cycle.

Bring Vixen to Heatmiser's on the Polar Express.

Bring Mrs. Claus to Vixen's slot by rotating Heatmiser's.

Have Mrs. Claus ride the Polar Express back to Snowmiser's to finish X.

Rotate Snowmiser's so that the Mayor shifts to where Mrs. Claus just arrived for Y.

Invert your X (3 moves).

Invert your Y (1 move).

Level 5 achievement unlocked?

Level 6

The set up for our next story is similar to what it was in Level 5.

    1.  Edge 1 Position & Orientation

        Vixen must share a plane with the Mayor and not Mrs. Claus.

        The color of one of her 2 tiles must match the color of the rest of that face.

            That face will become Snowmiser's.

    2.  Edge 2 Position

         The miserable Mayor must be at Snowmiser's with Vixen.

    3.  Edge 3 Position 

         After Vixen travels to Mothernature's equator, Mrs. Claus must be on the equator.

            Sometimes Mrs. Claus is across Vixen at Heatmiser's before Vixen travels.

            Sometimes she's waiting on the equator already.

    4.  Edge 3 Orientation

         After Vixen arrives to the equator, Mrs. Claus must be oriented to take her place.

            Vixen has two choices of travel; clockwise or counterclockwise on an outer plane.

            Mrs. Claus can only take Vixen's place by riding the Polar Express.

From a solved cube, set yourself up with:

(R^-1*MU²*R)*D^-1*(R^-1*MU²*R)*D

Then choose edges Vixen, Mrs. Claus, and the Mayor to cycle.

Bring Vixen to Mother Nature's equator in such a way that Mrs. Claus is correctly oriented.

    Vixen can choose to travel clockwise or counterclockwise on an outer plane.

    Only one of those two ways will correctly sets up Mrs. Claus.

Bring Mrs. Claus to Vixen's slot.

Invert your first move to finish X.

Rotate Snowmiser's so that the Mayor shifts to where Mrs. Claus just arrived for Y.

Invert your X (3 moves).

Invert your Y (1 move).

Level 6 achievement unlocked?

Level 7

On the other hand, let's explore the situation where we're thinking of two edges as being on a middle slice and one on a plane instead of the other way around.  As usual, Vixen wants to take the Mayor's slot, the Mayor wants to go to Heatmiser's, and Mrs. Claus wants to take Vixen's place.  When we begin,

    1.  Edge 1 Position & Orientation

        Vixen must be at Mother Nature's equator with the Mayor and not Mrs. Claus.

        Vixen must be oriented to arrive at the Mayor's slot correctly.

        Remember, Vixen travels on an equatorial Polar Express ride.

    2.  Edge 2 Position

         The Mayor is also on Mother Nature's equator.

    3.  Edge 3 Position

        After Vixen travels to Heatmiser's, Mrs. Claus must be waiting at Heatmiser's.

    4.  Edge 3 Orientation

        After Vixen arrives in Heatmiser's, Mrs. Claus must be correctly oriented to take her place.

            Vixen has two choices of travel; clockwise or counterclockwise on an outer plane.

            Mrs. Claus can only take Vixen's place with a Heatmiser rotation.

From a solved cube, set yourself up with:

(R^-1*D²*R)*(MU^-1)*(R^-1*D²*R)*(MU)

Once you've identified Vixen, Mrs. Claus, and the Mayor,

Bring Vixen out of the way into Heatmiser's with an outer plane rotation.

Vixen had two choices of travel, but only one where Mrs. Claus would be correctly oriented.

Rotate Mrs. Claus around Heatmiser's to take Vixen's place.

Invert your first rotation to bring Mrs. Claus to the equator.

That was X.

Have the Mayor ride the Polar Express to where Mrs. Claus just landed.

That was Y.

Finish the commutator, X^-1*Y^-1.

Level 7 achievement unlocked?

Those three feel different to me because of the placement of the M-movement, but in all cases, we used one M-move and its inversion.  Don't forget that you can use the inversion of any of these to change the direction of your 3-cycle.  Vixen is always the leader; whichever direction Vixen begins the 3-cycle is the direction Mrs. Claus and The Mayor will travel as well.

Remember, if your edges aren't set up in such a way that you're clear for takeoff, you can always use conjugates to get there.  Furthermore, the C for your conjugate can be made with M-moves.  Conjugates can be constructed in any way you like at all.  No rules.  Just right.

CHAPTER 13: THE EDGE 3-CYCLES PART II

You've made it past the first act of the edge 3-cycles, and second acts are always shorter.  Altogether, this post covers no fewer than five edge 3-cycles, and if you count their inverses, ten.  You're doing great.

Levels 8 and 9 are different than what we've done previously, because the intersection of X and Y in these commutators are 3-cubie strips instead of single cubies.  It appears that these new commutators permute three edge cubies without otherwise disturbing the rest of the cube, but that's not true!  They also reorient two center cubies by 180°.  Rubik's cubes with oriented centers are called supercubes.  If you find yourself solving a supercube and your centers are already in place, don't use these.  Make a conjugate with one of the edge commutators from chapter 12 instead.  On the other hand, if you're wanting to reorient center cubies and you don't care what's happening with your edges, these could be handy.  Level 8 reorients two adjacent center cubies, and level 9 reorients two opposite center cubies.

Level 8

Sometimes, Vixen, the Mayor, and Mrs. Claus are all at Mothernature's equator.  We want Vixen to be correctly oriented in one of Mothernature's lands, meaning that her tile will match that land, but she'll be on the opposite side of where she needs to be.  We'll call that land "Vixenland."

The set up is as follows:

    1.  Edge 1 Postion & Orientation

         Vixen, the Mayor, and Mrs. Claus are all on Mothernature's equator.

            The color of one of Vixen's 2 tiles must match the color of the rest of that face.

            That face will become Vixenland.

    2.  Edge 2 Position

         The miserable Mayor must be at Vixenland.

         (Vixenland, a province of Mothernature's, is also cold.)

            The Mayor has no Vixenland tiles or desire to be there.

    3.  Edge 3 Position 

         Mrs. Claus must be closer to Vixen than she is to the Mayor.

            So Vixen is in the middle, the Mayor is on one side, and Mrs. Claus is on the other.

            There is a solved edge cubie on the corner between Mrs. Claus and the Mayor.

    4.  Edge 3 Orientation

         Mrs. Claus loves Vixenland and has a Vixenland tile.

        Her Vixenland tile is not correctly oriented to travel to Vixenland equatorially.

Rotating Vixenland by 180° switches Vixen with the Mayor in one fell swoop.

(Notice that Vixen normally exchanges places with Mrs. Claus first.)

X is already finished!

Mrs. Claus rides the Polar Express into Vixenland, incorrectly oriented.

That was Y.

The Mayor and Mrs. Claus switch places in X^-1, correctly orienting Mrs. Claus.

The cube is fully restored with Y^-1.

Set yourself up with U² *MR^-1*U² *MR, identify your characters, and re-solve.

Level 8 achievement unlocked?

Level 9

But what if you have four edges to solve?  You can do this with two 3-cycles, of course, but sometimes, you'll find a related situation to the one above.  In this case, both Vixen and Mrs. Claus will be in Vixenland, correctly oriented with Vixenland colors, while both the Mayor and Ignatius Thistlewhite will be in Southtown (opposite Vixenland), correctly oriented with Southtown colors!  There'd be 4 guests in Mothernature's equator!

In this case, we'd exchange Vixen and Mrs. Claus by rotating Vixenland 180° for X, then rotate Mothernature's 180° for Y, and then finish our commutator.

Try to set this one up yourself, and repeat the algorithm to re-solve.

Level 9 achievement unlocked?

If, however, Vixen and Mrs. Claus ended up correctly oriented while the Mayor and Ignatius were not, we'd need something like a "2 edge flip" after that double 2 cycle.

CHAPTER 14: THE 2 EDGE FLIP

Just like we took a corner out, spun it around, and returned in chapter 11, we can do that with edges.  And as before, the laws of the cube dictate that if we reorient one, we must reorient another.

Level 10

If you'll recall, I was very generous with an oriented example in chapter 4.

So shall I continue introducing the reorienting commutators this way.  Lucky you!

It looks like I'm still right handed, so let's try to take out that UR edge cubie.

As before, let's bring it down with R^-1.

Now we need to move it out of the way with MU.

    [(MU)^-1 would get it out of the way, but make it harder to put back in.]

We'll mostly fix the top with R.

To get that cubie back to its place but in the wrong direction, we have to repeat R.

Now (MU)² will put that edge cubie back in place, but flipped.

We can bring it back to the top with R^-1.

This was the X in our commutator: R^-1*MU*R²*(MU)²*R^-1.

Y has to be some rotation of U, and U works.

Choosing U means we're choosing the counterclockwise edge cubie as our second one to flip.

Altogether, say it with me:

[R^-1*MU*R²*(MU)²*R^-1]*U*[R*(MU)²*R²*(MU)^-1*R]*U^-1

Did you flip two edges?

Returning to storyland, let's figure out what just happened.

Snowmiser's and Heatmiser's are any two opposite planes, and when our story begins, the cube is already solved with the exception of two edge cubies named Jingle Bells and Jangle Bells.  They are both occupying Snowmiser's.  They're in their correct slots, but require reorienting.

We pulled corner Jingle Bells out of Snowmiser's with a Vixenland rotation (why not) into Mothernature's equator.  From there, he rode the Polar Express clear off Vixenland.  Next, we rotated Vixenland twice, and when Jingle rode back in on the Polar Express, he was reoriented.  We restored Vixenland to its original position, and these 5 steps made up our X.

We rotated Snowmiser's to select our Jangle, and that was our Y.

We inverted our X, which reoriented Jingle and also solved what we had messed up.

Finally, we inverted our Y to straighten up Snowmiser's.

Re-solve your cube.

Level 10 achievement unlocked?

CHAPTER 15: THE ORIENTED 3-CYCLES

While I consider 3-cycles to be the meat and potatoes of the Rubik's cube, the cube is very often solved in layers.  If the first two layers are solved and the last layer has been arranged so that the cubies are correctly oriented, it's necessary to find permutations that preserve orientation.

The main thing to remember at this stage is that conjugates are like magic.  They can be any rotations and any number of rotations (No rules.  Just right.)  From a solved position, do anything you want to get your corners or edges so they're ready for takeoff; just write down what you're doing if it's long.  That's your N.  Then solve those 3 cubies the way you normally would with your 3-cycle.  Then invert your N, and there you have it.  It's nearly too good to be true.

Level 11

Let's start with a solved cube and try an oriented corner 3-cycle.

N*(X*Y*X^-1*Y^-1)*N^-1.

Remember that Vixen, the Mayor, and Mrs. Claus are corner cubies again.

To begin, assign them to the solved corners you wish to permute.

Vixen and the Mayor need to be at Snowmiser's, and they are.

Mrs. Claus, who's another corner at Snowmiser's, needs to be at Heatmiser's.

A 90° rotation of (any) Vixenland will accomplish this.

Choose one to put Mrs. Claus into Heatmiser's; that's your N.

Both Vixen and the Mayor will have remained correctly oriented in Snowmiser's.

Bring Vixen to Heatmiser's in such a way that Mrs. Claus is correctly oriented to take her place.

    Only one of the two Snowmiser corner cubies will work as Vixen.

    The plane of rotation in this step will be the opposite plane of what you chose for N.

    Vixen will end up diagonally across from Mrs. Claus on Heatmiser's.

Rotate Heatmiser's twice, switching Vixen and Mrs. Claus.

Bring Mrs. Claus back to Snowmiser's, and we've finished our X.

For our Y, rotate Snowmiser's to bring the Mayor to where Mrs. Claus just landed.

Then continue with X^-1; bringing the Mayor over to Heatmiser's the same way Mrs. Claus came.

    Rotate Heatmiser's twice and then bring Vixen back.

    Straighten out Snowmiser's for Y^-1.

Undo that first conjugate, and there's your oriented edge 3-cycle.

Did you do it?

If you did, and if you can re-solve your cube,

Level 11 achievement unlocked!??!

Level 12

Last one, for

    On the twelfth day of Christmas, a present from a friend

    A conjugate with two moves in the N

Nobody likes when the last question on an exam is the hardest one, so we'll do this together.

Let's do an oriented edge 3-cycle, but make a point to use more than one plane of rotation in our N.

First, we'll choose three edge cubies to cycle, and they'll all be correctly oriented on Snowmiser's.

If I choose an edge to move off of Snowmiser's with a 180° Vixenland rotation into Heatmiser's, and then rotate Heatmiser's by 90°, I can get 3 edge cubies on a middle slice, which brings us back to the situation from the beginning of chapter 13.  Of course, the "equator" would now be running between Vixenland and Southtown, but that doesn't matter, and anyone who has made it this far is a total pro at reorienting.

Start with a solved cube.

I just checked and it looks like I'm still right handed, so one example of this would be:

(R²*D^-1)*[U²*MR*U²*(MR)^-1]*(D*R²)

That seemed to cycle 3 oriented edge cubies clockwise when viewed from the top.

You still with me?

Can you figure out how to cycle them counterclockwise?

We could just take the inverse of what we just did, or try to rotate everything differently.

With (R²*D) as the new N for the conjugate instead of (R²*D^-1), we would:

(R²*D)*[U²*(MR)^-1*U²*(MR)]*(D^-1*R²)

Do you see what else in our algorithm, besides (R²*D^-1), had to change?

Did you notice why?  If so, congratulations.  You've officially become a

HIGH SCORER!!

CHAPTER 16: BACK TO THE BEGINNING - CROSS & BLOCKBUILDING

I sort of skipped all the beginning stuff, where you start moving cubies around (usually with simple conjugates) without writing anything down or constructing commutators.  Take a 3x3x3 cube, and try to make the red cross you see on first aid kits.  Make sure the cross edges match the 4 adjacent center colors they're touching.  If red is the center of your cross, you’ll be matching all the red plane's edges, and you'll be working with the 5 colors that aren't orange.

If that's easy, make any 1x2x2 square.  You'll be orienting and permuting 3 cubies and working around 3 center colors.  Once you get the hang of that, make a 2x2x2 cube, where you'll orient and permute 4 cubies, still working with 3 center colors.  If that gets easy, start making a full layer, 1x3x3, or a 2x2x3 block.  You don't really need a teacher for that, but if there's a good one, it's the original source.

CHAPTER 17: BACK TO THE BEGINNING - SIMPLE Z & Y COMMUTATORS

Meanwhile, let's return to the first commutator we discussed, F*U*F^-1*U^-1 from chapter 5.  If a commutator looks just like that, as short and sweet as possible with two adjacent face rotations both moving either clockwise or counterclockwise, then it will affect four corner cubies and three edge cubies in a "Z" configuration on the cube.  Let's call them "Z-commutators."

If, on the other hand, the commutator looks like this, F*U^-1*F^-1*U, as short and sweet as possible with two adjacent face rotations moving in opposite directions, then it will also affect four corner cubies and three edge cubies, but in a "Y" configuration on the cube.  Let's call them "Y-commutators."

Try either on a solved cube and see if you can spot the Z or the Y.

Z and Y commutators can be very helpful in blockbuilding.  I'll refer to "either Z commutators or Y commutators or both" as Z/Y commutators.

    The rest of the cubing community calls the Z commutator (R*U*R^-1*U^-1) the "sexy move."

    They also call the Y commutator (R^-1*F*R*F^-1) the "right sledgehammer,"

    the Y commutator (L*F^-1*L^-1*F) the "left sledgehammer,"

    the Y commutator (F*R^-1*F^-1*R) the "right hedgeslammer," and the

    the Y commutator (F^-1*L*F*L^-1) the "left hedgeslammer."

The 3-cycles we've been creating are more complicated commutators that yield simpler outcomes, and Z/Y commutators are simpler commutators that yield more complicated outcomes.  In this chapter, we're observing what these simple commutators do, and taking advantage of them to permute our pieces in a way that's convenient for us.

(Aside: the Skewb permutation puzzle relies on simple commutators that yield more complicated outcomes only, as it's impossible to construct more complicated commutators that yield simpler outcomes!)

Imagine you've solved Snowmiser's plane.  Mothernature's equator, then, already has four center cubies and 0-4 solved edges.  Perhaps you'd like to put an edge in there.  You'll have to break up a Snowmiser corner to do that, and knowing how to do that with Z/Y commutators can be handy.

One leg of the Z/Y shape will be the 3-cubie intersection between the Mothernature planes that contain the Snowmiser corner and edge slot we're discussing.  Both other legs will be 3-cubie Heatmiser edges, which is convenient, because we don't care how much they mess up the unsolved Heatmiser plane.

We need to use two Z/Y commutators to get that edge cubie in place.  The point of the first Z/Y commutator is to bring the Snowmiser corner into Heatmiser's in such a way that it's correctly oriented next to the edge cubie we have our eye on.  The point of the second commutator is to move the 2-cubie unit into place.

Find a Mothernature's equator edge that's hanging out at Heatmiser's; let's call it Blitzen.

(If there are none, which happens 1/16th of the time, just choose any edge cubie on Heatmiser's and pretend it's Blitzen.  That will knock out a real Mothernature's equator's edge, and then we'll have a real Blitzen.)

Blitzen wants to get cozy with that solved Snowmiser corner next to her slot, right?  Let's call that corner cubie Carrot.  Blitzen has two colors, both of which belong in Mothernature's, but she's oriented on Heatmiser's in such a way that only one of those colors can touch a Mothernature center cubie and match colors.  Rotate Heatmiser's to make that happen; this is our opening position.

The first move in our first Z/Y commutator is the 90° Heatmiser rotation moving Blitzen in the direction away from Carrot.  The second move is the 90° Mothernature rotation that brings Carrot into Heatmiser's and doesn't disturb Blitzen.  Once the two inverses are applied to complete the commutator, lo and behold, Blitzen has her Carrot!

The first move in our second Z/Y commutator is a 90° Heatmiser rotation moving Blitzen and her Carrot in the direction of Blitzen's Snowmiser tile.  The second move is a 90° Mothernature rotation that doesn't disturb Blitzen and Carrot.  Once the two inverses are applied to complete the commutator, lo and behold, Blitzen and Carrot are right where they always wanted to be.  But this is hardly surprising, because you learned how to do this when you built your first 2x2 square and placed your first corner.

CHAPTER 18: 2 WRONG EDGES & 2 WRONG CORNERS I

If you've solved nearly all of your cube without a care in the world, you might stumble upon a situation in which everything is solved except for two edge cubies and two corner cubies.

Let's focus on the corners cubies first.

Say we have corners A, B, C, and D, and our corner slots look like this:

1 2

4 3

A belongs in 1, B belongs in 2, C belongs in 3, and D belongs in 4, so we want:

AB

DC

The two solved corner cubies will either be adjacent or across from one another.

When they're adjacent like this:

AB

CD,

then a single 90° rotation reduces the number of correctly placed corner cubies from 2 to 1.

AB → CA

CD     DB

This is progress, because 3 wrong corner cubies can be cycled into place!

Once they're in position, if they need to be reoriented, there's always the 2 corner flip.

When the two solved corner cubies are across from one another, however, that's worse.

The 4 rotations of the plane will never result in 1 solve, but 0 or 2 instead:

AD → BA → CB → DC

BC     CD     DA     AB

2        0        2       0

However, a 3-cycle will get the corner cubies into the adjacent stage.

AD → AB

BC     CD

From there, they'd need a rotation and another 3-cycle just as before.

All of that is also true for the edges, so there's no need to write it out twice.

Un-solving cubies to set up 3-cycles wasn't obvious to me, so I wrote to Jamie Mulholland asking what to do, and he sorted me out within the hour.

CHAPTER 19: 2 WRONG EDGES & 2 WRONG CORNERS II

Of course, this whole situation could be avoided in the first place by solving either all of the edges or all of the corners a little earlier in the process, for according to the laws of the cube, we couldn't have an issue like that with either set on their own.  

The blockbuilding methods often leave an unsolved 3-cubie column, which serves as an open path, perpendicular to the last unsolved face in order to orient and solve the last 5 edges using a subgroup of Group Rubik.  For example, taking the three faces we can see most easily, the unsolved face would be U, the cube's edge shared by R and F would be the open path, and the subgroup of rotations we'd create to orient the remaining unsolved edges would have elements (U, U², U^-1, R, R^-1, F, F^-1).  Transferring edges in and out of the open path with (R, R^-1, F, F^-1) can orient them to your liking, and the U rotations can place and store the oriented edges.  Gathering correctly oriented edges and storing them in the U layer must be what it feels like to be a squirrel hiding acorns.  Exactly two of the four U edge cubies must be solved relative to one another first, and then the remaining three edge cubies (two in U and one in the open path) can be solved all at once.  At this point, when all the edges are re-solved with the open path or commutators, you're home free with the corner 3-cycle, which should be the trustiest tool in your box.

However, if you're smarter than I am (I tend to hover around the squirrel level), you can break out of 3-cycles on single cubies for a moment to permute both edge and corner cubies together.  Explained by Michael Hutchings, let's say Z is a Z-commutator.  If we can think of a rotation that keeps all of these cubies in one layer, we can consider that to be the N^-1 for the conjugate, N*Z*N^-1.  It follows that N*Z*N^-1 would be an algorithm that permutes both edge and corner cubies within a single layer.  In the example F*U*F^-1*U^-1, either L^-1 could bring all of the affected cubies to the U face, or R^-1  could bring them to the F face.  So both L*Z*L^-1 and R*Z*R^-1 would be single layer algorithms that permute both edge and corner cubies:

L*(F*U*F^-1*U^-1)*L^-1 puts all affected cubies are in the U face;

R*(F*U*F^-1*U^-1)*R^-1 puts them in F.

Also, if you've practiced the corner 3-cycle enough, you can extend that method to pairs of cubies instead of single cubies, and pairs of cubies would reorient and permute both corner and edge cubies together.  That is super cool but not often relevant.  The original source is probably necessary to elaborate on that concept.

CHAPTER 20: CONCLUDING REMARKS & SOURCES

There were two major points I was aiming to make with this post.

The first is that once you understand how all this works, you can create your own solutions even if you don't do them in the most efficient way.  If you're doing a 2-corner twist, it's okay if you take that corner out and restore that face with more than 3 rotations for half of your X.  It's okay if you put it back with more than 3 rotations.  Just remember what you did or write it down.  The same is true with conjugates; it's okay if you don't find the shortest one, as long as you can apply its inverse at the end.  We don't have to be fast, and we don't even have to be efficient.

The second is that if you can understand how these commutators and conjugates work, you can identify your Vixens or your Jingles, and you don't have to be glued to oriented algorithms.  There's nothing wrong with rotating the cube and letting muscle memory help you, but it's also nice to have the power of freedom.

If and only if you unlocked ACHIEVEMENTS 1-10, congratulations!  You beat the boss!

Do you believe in yourself?  Do you believe in the Rubik's cube?

Click here to view the end of our game.

SOURCES

ryanheise

jamiemulholland

michaelhutchings

chrishardwick

michaelgottlieb

PS.  The next post is even less fun!

Entire Series:

    The 4x4x4 (2/5)

    Mommy's Permutation Puzzle Class (3/5)

    The Skewb & SQ1 (4/5)

    Putting It All Together (5/5)

Rubik's Cube

Snow Miser, Mother Nature, and Heat Miser

Jingle Bells, Vixen, & Jangle Bells

The Mayor of Southtown

Mrs. Claus

Mr. Thistlewhite, Mrs. Thistlewhite, & Ignatius Thistlewhite