The Skewb & SQ1 (4/5)

INTRO

This is my fourth post on permutation puzzles.  The others are here:

    The Rubik's Cube (1/5)

    The 4x4x4 (2/5)

    Mommy's Permutation Puzzle Class (3/5)

    Putting It All Together (5/5)

The first post covers conjugates and commutators and how they pertain to the Rubik's Cube, the second covers the notion of a "parity problem," and the third documents my kids' course.  From developing an interest in these puzzles, I've acquired the following collection:

15 Puzzle

Tower Cuboid, Domino Cuboid

Ivy Skewb, Dino Skewb, Skewb

Pocket Cube, Rubik's Cube, Rubik's Revenge, Professor's Cube

Cylinder Shape Mod, Windmill Shape Mod, Fisher Shape Mod, Axis Shape Mod

Pyraminx, Pyramorphix, Mastermorphix

Kilominx, Megaminx

Square 1

20 Stickerless QiYi Brand Permutation Puzzles

Sleepin' in Their Basket

The reasoning behind solving all of these puzzles is related, but the Skewb and the Square 1 (SQ1) seem to require some additional attention.  That, and the fact that my faithful friend Jack keeps editing these posts out of the goodness of his heart, is why I'm back.  I'll try to make it short.

Skewb, SQ1, Domino Cuboid

CHAPTER 1: THE SKEWB

Chapter 17 of my original Rubik's Cube post reads:

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!)

The Skewb has 6 centers and 8 corners.  One twist on the Skewb displaces half the puzzle: 3 centers and 4 corners.  It's like the Pocket Cube in terms of severe displacement, but without the option to create simple X's and Y's that that have an intersection of 1.

If we can't easily create 3-cycles, we must try a (simple) commutator X*Y*X^-1*Y^-1 in which both X and Y are two different 120° rotations, and make a note of the results.  Such a commutator on the Skewb preserves 6 pieces: 4 corners surrounding a center, that center, and an adjacent center.  The remaining 4 centers have been permuted by two 2-cycles, and the remaining 4 corners have been reoriented.  Because of the two 2-cycles, doubling the commutator, X*Y*X^-1*Y^-1*X*Y*X^-1*Y^-1, is an algorithm that affects corner orientation only!

Thus, the way the Skewb is solved is as follows:

One center and its surrounding four corners are solved by blockbuilding.

The corners must coordinate with one another, but not with other centers.

This automatically places the final four corners.

Let X and Y be 120° rotations that do not disturb the solved center.

The commutator X*Y*X^-1*Y^-1 is then used to place the remaining centers.

Finally, doubling the commutator, X*Y*X^-1*Y^-1*X*Y*X^-1*Y^-1, reorients the final four corners.

The Skewb has to be correctly positioned for this to work, but I'm sure you can figure that part out much more easily than if I write another paragraph about it.  Doing it wrong won't permute anything anyway, so you don't have much to lose.

Although the initial reasoning behind this is a little tricky, solving the cube becomes very easy rather quickly.  It's a lovable little thing.

CHAPTER 2: SQ1 INTRO & NOTATION

If the Skewb is a lovable little thing, SQ1 is quite the opposite.  It's nerdy.  And for an official "World Cube Association" event, unpopular.  Maybe even annoying.  It's so unlike its peers.

The first thing to notice is that it's structured differently.  SQ1 is a 3 layer cube in which the middle layer is a 2-piece equator that is very easy to manipulate.  Solving the puzzle, then, is really a matter of solving the top and bottom layers.  I'll call these U and D.

SQ1 is a shape mod that's first made into a cube.  Typically, the yellow and white pieces are placed into their respective sides, and then everything gets permuted around.  At some point in this process, the parity state must be addressed.  More on that in a mo'.

Because the SQ1 moves so differently, it requires its own notation.

To read notation, it is assumed that

    the shorter part of the equator when viewed from the front is always on the left.

The equator only allows 180° turns, so the notation   /   means a 180° turn on the right.

Meanwhile, measuring angles from the center of U or D, each corner piece is 60°.

Each edge piece is 30°.

    (Each of 12 corner and edge pieces on a SQ2 are 30°!)

U and D rotations are therefore based on the number of 30° angles.

As is standard, a positive number indicates clockwise, and negative indicates counterclockwise.

Putting this together, 3 = 90° clockwise.

5 = 150° clockwise.

-3 = 90° counterclockwise.

() are used to organize numbers to mean a rotation on U and a rotation on D.

The notation (3,3) means a 90° clockwise U turn and a 90° clockwise D turn.

Remember, clockwise is determined "when looking at the face."

So (3,3) requires that U and D faces to shift in opposite directions.

(1,-1) means a 30° clockwise U turn and a 30° counterclockwise D turn.

U and D would then move in the same direction, imitating a middle slice move.

(1,6) means a 30° clockwise U turn and a 180° D turn.

Putting this together with 180° R turns (/), an example of SQ1 notation is:

    (6,0)/(6,0)/(6,0)/(6,0)

This algorithm flips the right side of the equator.

For all algorithms, the left side of the equator never moves.

Another example of SQ1 notation is:

    /(6,6)/(-1,1)

That one exchanges U with D.

If an algorithm affects either U or D only, knowing how to exchange U and D can be very handy.  For example, let's say that I want to permute something and my algorithm affects U only, but my issue is in D.  I can bring D to U, perform my algorithm, and swap them back.  Conjugates :)

CHAPTER 3: SQ1 CUBE SHAPE

The first step is to get this thing looking like a cube.  To do that, we need to get 4 corners alternating with 4 edges in both U and D.  Intuitively, this is not one of our skills; our hunting and gathering backgrounds did nothing for us here.  One thing we can do, however, is recognize symmetry.  The most straightforward method to go about cube shape is to find symmetry.

Fuss about until all 8 adjacent edges and 2 corners are in U, leaving D with 6 corners.  Of course you could do this backwards and put the 6 corners in U instead, but you wouldn't be able to see what you're doing, so I wouldn't recommend it.  Tingman calls that shape "the Millennium Falcon," and I think that's a pretty good name for it.  After we find the Millennium, we split the 8 edges into two 4's, the two 4's into four 2's, four 2's into four 1's and two 2's, and then finally into all eight 1's.  Then it's a cube.

Millennium Falcons in Pretty Colors

Once the SQ1 is in cube shape, if U or D (not both) is offset by exactly 1, then / maintains cube shape.  Of course, 3's, -3's, and 6's after (1,0) or (0,-1) won't destroy cube shape either.  It's handy to know this when exploring your own algorithms.  

For example, the algorithm (1,0)/(-1,0) exchanges half of U with half of D.

Try it and its inverse.

Here's the right equator flip we saw in the last chapter, but without leaving cube shape.

    (1,0)/(6,0)/(6,0)/(5,0)

        Let's call it the "Equator Flip."

        Repeating it will restore the cube.

Here's the U and D flip without leaving cube shape.

    (1,0)/(6,6)/(-1,0)

        For all you Stranger Things fans, let's call it the "Upside Down."

        Repeating it will restore the cube.

    (1,0)/(-1,-1)/(0,1)

        transposes the pair of front and back edges of U and D.

        Because those edges gallop around so fast, let's call it the "4 Horsemen."

        Repeating it will restore the cube.

    (1,0)/(3,0)/(-1,0)

        swaps a quarter of U with a quarter of D;

        It also shifts a quarter of U and D clockwise.

        Interesting!

So (1,0) puts us in cube shape position and /(3,0)/ starts permuting quarter squares (quarters).

Playing with that, it appears that

    (1,0)/(3,0)/(3,0)/(3,0)/(3,0)/(3,0)/(2,0)

        cycles quarters around; the whole thing is an identity algorithm.

CHAPTER 4: SQ1 PARITY PROBLEMS

It's time to talk about parity.

Imagine there are 3 corners on the right side of U and 3 corners on the right side of D.  It's true that a / creates 3 2-cycles of corners.  A 2-cycle is a transposition, so that's an odd number of transpositions!!  The entire point of my second post was to explain that an odd number of transpositions is... unkind.  The Rubik's Cube never shows such unkindness.

Remember how when we were getting into cube shape, we put 6 corners into D?  Well, it's also true rotating the 6 corners once, like (0,2), creates a 6-cycle of corners, which requires 5 transpositions.  That's an odd number of transpositions again!  Unkindness again!

(The SQ2 looks like a tremendous pain, but at least it does not have these problems.  A / would create 6 transpositions instead of 3, and a (0,2) would create 10 transpositions instead of 5.  But I digress.)

So there are parity problems in SQ1, and two ways we know to wrong the right or right the wrong.

Advanced SQ1 competitors know how to check for parity problems before getting into cube shape.  While that's ideal, it requires memorizing a "reference scheme" and making a bunch of calculations concerning 6 different things.  I'm not against calculations, as ridiculous as they may be, but I'll be damned if I'm memorizing a "reference scheme."

Other cubers nearly solve the SQ1, find there is parity half the time, curse, and then apply some monstrous algorithm at the very end.  That is also entirely unreasonable.

What to do, what to do.  Well.

Because I refuse to memorize a reference scheme, the first step would have to be for me to get this thing into cube shape.  After that, instead of orienting or permuting pieces, I could start counting all the cycles for the corners and the edges as a way of checking the parity of all transpositions required to solve the cube.  Do you remember when I was talking about subgroups under chapter 3 on group theory in my first post?  Of course not.  That part goes over counting cycles.

Let's say I begin my parity checking process by counting a corner cycle.  At random I'd choose a corner, C₁, which is in a slot, S₁.  If C₁ is solved, meaning that S₁ is C₁'s correct slot, then we can say C₁ is part of a 1-cycle and it requires 0 transpositions to become solved, and that cycle's parity is even.

If C₁ is not solved, meaning that S₁ is not C₁'s correct slot, I'd look at C₁'s correct slot to find a new corner, C₂.  If C₂'s correct slot was S₁, then C₁ and C₂ would make a 2-cycle, which requires 1 transposition, and that cycle's parity would be odd.

But if C₂'s correct slot was not S₁, I would look at C₂'s correct slot to find C₃.  If C₃'s correct slot was S₁, then C₁, C₂, and C₃ would make a 3 cycle, which requires 2 transpositions, and that cycle's parity would be even.

And so on and so forth.

I'd add up the number of transpositions of all the cycles for corners and all the cycles for edges on U and D to determine the overall parity state.  If the cube happened to be completely solved, that number would be 0.  If all 8 corners were involved in a single cycle, that would require 7 transpositions, and if all 8 edges were involved in another single cycle, that would require 7 more.  In other words, I must count to some number ≥ 0 but ≤ 14.

Once the puzzle is in cube shape and remains in cube shape, the parity of transpositions is locked.  To avoid parity problems, we want the cube shape to be locked in an even parity.

If the parity of total transpositions were even, I would simply continue solving the cube.  If it were odd, I'd have to fix that by taking my cube back to the Millennium with /(3,3)/(1,2)/(4,2)/.  Then I'd (0,2) to swap parity, and restore to cube shape once again.

Altogether, that would look like this:

         /(3,3)/(1,2)/(4,2)/

         (0,2)

         /(-2,4)/(1,2)/(-3,-3)/

But wait!  We know of not one, but two ways to right the wrong!  I could also toggle parity by taking my cube back only most of the way towards the Millennium, /(3,3)/(1,2)/, noticing an abundance of adjacent corners on both my U and D.  Then I could move the 3 adjacent corners on U and and the 3 adjacent corners on D over to the right side with (2,-2), and swap parity with /.  Then I'd restore cube shape.  That would be kind of showoffy.

Altogether, that would look like this:

        /(3,3)/(1,2)/

        (2,-2)/

        (4,-4)/(1,2)/(-3,-3)/

CHAPTER 5: BUILDING SQ1 ALGORITHMS

By now, you may have wondered why I included the Domino Cuboid in my photo.  To solve the Domino Cuboid in terms that made sense to me, I used simple commutator/blockbuilding skills to simultaneously solve the edges, and then I wrote a commutator for the corners.  The X in my commutator was R²*U*R²*U^-1*R², and the Y was some form of D.  Mainly, my point is that the R² reminds me a lot of the /.  The 180° flip is weirdly constricting, but solving the Domino gives me the courage to explore the SQ1.  I recommend trying these puzzles in that order.  With the notion that we're going to be using something like R²'s all the time, let's start building algorithms.

Here's where we left off at the end of chapter 3:

    (1,0)/(3,0)/(-1,0)

        swaps a quarter of U with a quarter of D;

        it also shifts a quarter of each U and D clockwise.

        This is an algorithm about making quarters, so let's call it the "Washington."

        And if we're going to be talking about quarters, let's abbreviate them.

        FR - front right quarter

        FL - front left quarter

        BR - back right quarter

        BL - back left quarter

Say we wanted to permute quarters but maintain orientation.

We know

    (1,0)/(3,0)/ would get us started.

    (3,3)/(3,0)/ would move all the quarters to their original faces.

    (5,6) would reorient the cube so we could see what happened.

Altogether,

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6)

        swaps BR & BL in both U and D.

        This is an algorithm about 4 oriented quarters, so let's call it "2 Heads 2 Tails."

        Repeating it will restore the cube.

Also, because

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6)

        swaps BR & BL in both U and D,

I should be able to perform the algorithm, reposition one face only, and repeat the algorithm.

The second iteration will re-solve the face I didn't reposition!

That's just like what I was talking about with the Skewb.

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

        Aha!

        This one swaps FL & BL in D only.

        Repeating the whole thing will restore the cube.

        This is an algorithm about two oriented quarters only, so let's call it the "Half Dollar."

        But actually, "Kennedy" is cooler.

        Repeating it will restore the cube.

Related Kennedys:

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,-3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

        swaps FR & BR in D only.

        Repeating it will restore the cube.

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,6) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,3)

        swaps FL & BR in D only.

        Repeating it will restore the cube.

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (3,0) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(-2,6)

        swaps FR & BR in U only.

        Repeating it will restore the cube.

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (-3,0) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(-1,6)

        swaps FL & BL in U only.

        Repeating it will restore the cube.

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (6,0) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(2,6)

        swaps FR & BL in U only.

        Repeating it will restore the cube.

Let's go back to Washington.

     (1,0)/(3,0)/(-1,0)

        Specifically, this moves

            BL in U to FR in D

            BR in D to FL in U

            FL in U to BL in U

            FR in D to BR in D

        Washington doesn't affect FR & BR in U or FL & BL in D.

That means I should be able to

    1.  Flip 4 quarters around with Washington,

    2.  Reposition U and D with (3, 3) so the scrambled quarters are in the back, and

    3.  Swap adjacent quarters with 2 Heads 2 Tails to get one unsolved quarter on each face.

           Let's try it out:

    (1,0)/(3,0)/(-1,0) (no slash, just resetting)

    (3, 3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(2,3)

        Eureka, it worked!!!

        BL in U transposed with BR in D.

        Repeating it restores the cube, natural, natural.

        This is an algorithm that maintains 3/4 of each face!

        Let's add up those quarters and celebrate with some "Charles Shaw."

I was hoping to get to something like this.  Originally, I was thinking that from here I'd further isolate a corner or an edge to make a commutator.  But as I'm entirely unwilling to leave cube shape even for a moment, all I can permute are quarter squares.  Also, so many of my algorithms are quarter square transpositions that leave the rest of the cube intact, which means I've already acquired more powerful tools than I had predicted.  So it looks like I need to reevaluate my plan and solve this thing by reduction!

CHAPTER 6: THE DOUBLE KENNEDY

Well, to be honest, I failed.  I was able to solve my SQ1, but not in a way that I was happy to present to you.  I still had to derive one more thing.  Begrudgingly, I admit that it appears to be important to understand both types of quarter groupings: (1,0) and (0,-1).  Luckily for us, that's not difficult at all.

There comes a time in blockbuilding quarters where you can run into something that feels like a parity error due to the limitations of my algorithms.  So we need one more tool.  You see, our quarter swapping algorithms make a 2-cycle of corners and a 2-cycle of edges.  The 4 Horsemen makes one 2-cycle of edges in U, and another in D.  But what we can't do yet is make 2 corner or edge transpositions in only one face.  That's also known as a 3-cycle, and those are all over my other permutation puzzles.  It took me a little while to figure this one out, but I finally got it.

First, let's just explore what (1,0) and (0,-1) quarters are in the first place.

When using algorithms that begin with (1,0) and not (0,-1),

    Quarters are edges then corners when moving clockwise.

    Think of corners as having 2 colors, not 3.

    The first corner color is yellow or white.

    The second edge-matching color is on each corner's counterclockwise side.

    The third color is completely ignored.

    We'll call these (1,0) quarters.

On the other hand, when using algorithms that begin with (0,-1) and not (1,0),

    Quarters are corners then edges when moving clockwise.

    The first corner color is yellow or white.

    The second edge-matching color is on each corner's clockwise side.

    We'll call these (0,-1) quarters.

    Because our algorithms were written for (1,0) quarters, we mostly use those.

    Luckily, extending our understanding to (0,-1) quarters is not hard at all!

If we momentarily think in (0,-1) quarters and then return to (1,0) quarters, we can move edges around but leave corners untouched!  But we never have to stop thinking in quarters!  We just have to think in two kinds of quarters with overlapping corners!

A (0,-1) Kennedy looks like this:

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,-5) (no slash, just resetting)

    (0,3) (rotating)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,1)                 

        That maintains U and swaps FL and BL (0,-1) quarters in D.

To turn that into a edge 3-cycle, follow up with the corresponding (1,0) Kennedy!

We started with a (0,-1) Kennedy DL, so in this case, we'll continue with a (1,0) Kennedy DL:

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)   

        Look!  It's a clockwise 3-cycle of D front, back, and left edges!  Thrilling!!

I'll call the 3-edge cycle a DoubleKennedy.

Specifically, that one was the DoubleKennedyDL, and altogether, it looks like this:

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,-5) (no slash, just resetting)

    (0,3) (rotating)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,1)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

        Cycles D front, back, and left edges clockwise

DoubleKennedy DR:

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,-5) (no slash, just resetting)

    (0,-3) (rotating)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,1)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,-3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

        Cycles D front, back, and right edges clockwise

Counterclockwise DoubleKennedys, on the other hand, begin with (1,0) Kennedys:

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,-5) (no slash, just resetting)

    (0,3) (rotating)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,1)

        Cycles D front, back, and left edges counterclockwise

    (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (no slash, just resetting)

    (0,-3) (rotating)

    (1,0)/(3,0)/(3,3)/(3,0)/(5,0)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,-5) (no slash, just resetting)

    (0,-3) (rotating)

    (0,-1)/(3,0)/(3,3)/(3,0)/(6,1)

        Cycles D front, back, and right edges counterclockwise

You get the idea.  You could cycle edges in U by changing (0,3) to (3,0), etc.

I won't muddy up our toolbox with all of these because you don't actually need to understand it for my method.  You just need to be able to start an algorithm with (0,-1), like maybe one time.  Of course, now that you know how to make 3-cycles of edges, you can solve a SQ1 a great variety of ways.

CHAPTER 7: SOLVING THE SQ1

  7.1: TOOLBOX OF DERIVABLE ALGORITHMS  

Dolla'Dolla': (1,0)/(-1,0) (Chapter 3)

Equator Flip: (1,0)/(6,0)/(6,0)/(5,0) (Chapter 3)

Upside Down: (1,0)/(6,6)/(-1,0) (Chapter 3)

4 Horsemen: (1,0)/(-1,-1)/(0,1) (Chapter 3)

Washington: (1,0)/(3,0)/(-1,0) (Chapter 5)

2 Heads 2 Tails: (1,0)/(3,0)/(3,3)/(3,0)/(5,6) (Chapter 5)

Kennedy DR: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(0,-3)(1,0)/(3,0)/(3,3)/(3,0)/(5,0) (Chapter 5)

2H2T(0,-3)2H2T

Kennedy DL: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(0,3)(1,0)/(3,0)/(3,3)/(3,0)/(5,0) (Chapter 5)

2H2T(0,3)2H2T

Kennedy D Opposite: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(0,6)(1,0)/(3,0)/(3,3)/(3,0)/(5,3) (Chapter 5)

2H2T(0,6)2H2T

Kennedy UL: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(-3,0)(1,0)/(3,0)/(3,3)/(3,0)/(-1,6) (Chapter 5)

2H2T(-3,0)2H2T

Kennedy UR: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(3,0)(1,0)/(3,0)/(3,3)/(3,0)/(-1,6) (Chapter 5)

2H2T(3,0)2H2T

Kennedy U Opposite: (1,0)/(3,0)/(3,3)/(3,0)/(5,6)(6,0)(1,0)/(3,0)/(3,3)/(3,0)/(2,6) (Chapter 5)

2H2T(6,0)2H2T

Charles Shaw: (1,0)/(3,0)/(-1/0)(3,3)(1,0)/(3,0)/(3,3)/(3,0)/(2,3) (Chapter 5)

WASH(3,3)2H2T

Parity Rotation: /(3,3)/(1,2)/(4,2)/(0,2)/(-2,4)/(1,2)/(-3,-3)/ (Chapter 4)

Parity Flip: /(3,3)/(1,2)/(2,-2)/(4,-4)/(1,2)/(-3,-3)/ (Chapter 4)

  7.2: THE KRATZKE METHOD  

    1.  Get to the Millennium Falcon - Tingman's video is excellent for this.

        a.  Group pairs of edges together.

        b.  There are 3 ways to add the final pair to the cluster of 6.

            I.  They can be added to a side of the cluster of 6.

            II.  They can be separated into a line and added to straddle the cluster of 6.

            III.  If the last 2 edges are stuck in an L-shape instead,

                Replace an outer pair in the cluster of 6 with the L.

                This creates a line of edges, a cluster of 4, and an isolated pair.

                Make a cluster of 6, and then add the line.

        c.  The notion of opposite edges in a line is important; let's call these edge-lines.

    2.  Do the carbon dating thing to get a cube.

    3.  Check for odd parity.

        a.  You can rotate pieces around as much as you like before counting cycles.

        b.  Once you start counting cycles, however, do not alter the cube.

        c.  While unnecessary, it's easier count cycles if the pieces are oriented.

            That means all the yellow pieces are in U and all the white pieces are in D.

        d.  To do this, first put yellow corners into U with Dolla'Dolla'.

                Whenever I specify an algorithm like that, I always mean:

                You can use it as many times as needed, and

                You have the freedom to use U and D rotations to your heart's content.

            Then use the 4 Horsemen to get one or both yellow edge-lines into U.

                It's possible that this leaves an L-shape of edges in white.

                In this case, permute white corners to form yellow and white edge-lines.

                Whenever I say "permute," feel free to use any of our derivable algorithms.

            Finally, use the 4 Horsemen to finish orienting edges.

        e.  Calculate the parity of transpositions needed to solve all pieces.

    4.  If the parity is odd, use one of the parity fix algorithms.

    5.  Blockbuild exactly 4 quarters and hide them from the 4 Horsemen.

        Then build the remaining 4 quarters with the 4 Horsemen.

        You must be hiding 4 (1,0) quarters or 4 (0,-1) quarters; don't mix and match.

        a.  Odd numbers of built quarters are a huge pain; avoid them if at all possible.

        b.  (1,0) quarters can be hidden in FR and BL of U and also FL and BR of D.

            (0,-1) quarters can ben hidden in the opposite quadrants.

            Built edges simply need to be perpendicular to the 4 Horsemen transpositions.

        c.  Make U into yellow corners with white edges.

            (You can choose to divide pieces any way you like; this is just how I do it.)

            (Ex: white corners with yellow edges; hot color corners with cool color edges)

            Arrange U with the method I described before the parity count in 3d.

        d.  Permute quarters in D until the 4 Horsemen can build 2 opposite yellow quarters in U.

        e.  Build the 2 yellow quarters and hide them from the 4 Horsemen with a 90° U rotation.

        f.  If you're very lucky,

            I.  You didn't accidentally build any white quarters, AND

            II.  You're set up so that the 4 Horsemen can build 2 white quarters in D.

                In this case, build those 2 white quarters!

                Then use the 4 Horsemen to finish building all 8 quarters.

        g.  Otherwise, permute quarters in D and build the 2 remaining yellow quarters in U.

            We must now evaluate D.

            I.  If D has 0 built quarters, excellent!!  We can hide them from the 4 Horsemen.

                Begin with Dolla'Dolla', follow with 2 Heads 2 Tails, and hide.

                Use the 4 Horsemen to build all four white quarters.

            II.  If D has 1 built quarter, we must remedy this with a (0,-1) Kennedy.

                We don't care about quarter placement, so we don't need a DoubleKennedy.

                (A (0,3) rotation cycles D front, back, and left edges clockwise.)

                (A (0,-3) rotation cycles D front, back, and right edges clockwise.)

                You have two options.

                    α.  Unbuild all white quarters with a (0,-1) Kennedy and go to the previous step.

                    β.  Solve U completely with (1,0) quarter algorithms first.

                        Then use your (0,-1) Kennedy to help make all 8 (0,-1) quarters!

                        That commits you to (0,-1) algorithms through the end.  Fancy.

            III.  If D has 2 built quarters, you counted wrong and you have parity.

            IV.  D can't possibly have 3 built quarters; what would be in the 4th?

            V.  If D has 4 built quarters, you already have all 8!  Go go go!

    7.  Permute your 8 quarters to finish the solve!!

CHAPTER 8: CHECKERBOARDING & SOURCES

You've made it all the way through my final post, and you deserve a paragraph on checkerboarding.  It's not important, and it's hardly any sort of concluding remark.  It's just something fun.

To make a solved standard Rubik's Cube into one with a checkerboard pattern on each side is very easy.  Take each of the 3 middle slices, and in any order, rotate them 180°.  That's all.

Checkerboarding doesn't entirely work with puzzles that have an even number of pieces along at least one edge, but some people try.  Just like the book Will It Waffle, here are my opinions on Will It Checkerboard.

15 Puzzle: No.  What does this even mean?

Tower & Domino Cuboids: Only on some sides; still worth it

Ivy Skewb, Dino Skewb, Skewb: Not at all

Rubik's Cube, Professor's Cube: Classic checkerboard!

Pocket Cube, Rubik's Revenge: I don't like this

Cylinder Shape Mod: Yes, kind of sweet

Windmill Shape Mod: YES; this one is named for its checkerboard pattern

Fisher Shape Mod: Yes, it's wacky

Axis Shape Mod: Yes, yikes, the wackiest

Pyraminx, Pyramorphix: No

Mastermorphix: Looks like bowling shoes

Kilominx, Megaminx: Just gets scrambled really

Square 1: So glad you asked

We should all know how to checkerboard a SQ1!

    4 Horsemen: (1,0)/(-1,-1)/(0,1) (no slash, just resetting)

                        (3,3) (rotating)

                        (1,0)/(-1,-1)/(0,1) (no slash, just resetting)

                        (-3,-3) (rotating)

Altogether, (1,0)/(-1,-1)/(0,1)(3,3)(1,0)/(-1,-1)/(-3,-2)

SOURCES

tingman

davidwoner

andyklise

kewbz.co.uk

brandonlin

cubemaster

Mommy's Permutation Puzzle Class (3/5)

Having written my courses on the Rubik's Cube and the 4x4x4 (final posts here and here), I was pleased to find that I had also graduated.  Imagine my astonishment when I realized that my diploma came with a teaching certificate! The fine print clarified that I was only qualified to teach my own children, but it was still a pleasant surprise, as I hadn't been expecting a teaching certificate at all.

Cuddly puzzle time with Mommy is when I take the concepts we just talked about and demonstrate them on the cube.  If you're trying to follow along (even though I'm not qualified to teach you) and you run into one of these, just zoom me and we can try it without the cuddles.  I don't know if that works.  I only know the cuddling way works.

The Cuddlers

CHAPTER 1: THE FIFTEEN PUZZLE

Hi babies!  Come to Mama.

Are you ready to learn how to play with Mama's toys?

("Yes!" with lots of hugs.  A moment to get comfortable and snuggly - Milli always needs extra time.)

Imagine you have a big bag of marbles in all your favorite colors!  Your best friend has a birthday party in an hour, and we forgot to get a present, so you'll have to pick out some of your marbles and put them into a birthday bag to make a birthday present, okay?  Did you imagine it?  Did you pick them?

In the world of math, you just found a "combination."  Any selection of a new group of marbles from your big bag of marbles is a combination.

Now let's pretend that your best friend's birthday party just got cancelled because of covid, and you get to keep all of the marbles after all.  (We can get something else for your best friend later.)  Let's take all the marbles from the birthday bag and dump them onto the floor and make them into a caterpillar.  When we're building a caterpillar, we're making an arrangement instead of a selection.

In the world of math, an arrangement is a "permutation."  We could scramble up your caterpillar and then make a new caterpillar from the same marbles!

Would we be making a selection or an arrangement?

An arrangement!

Would we be making another combination or another permutation?

A permutation!!

That's right.  In fact, it would be the same combination as before!

A combination is a selection of things where the order/placement/position does not matter

A permutation is an arrangement of things where the order/placement/position does matter

Combination questions usually sound something like, "How many different groups of three marbles can we make make from the big bag of marbles?"

Permutation questions usually sound something like "How many different caterpillars can we make from the caterpillar on the floor?"

All our toys have things to rearrange, so our toys are called permutation puzzles.

Our first toy to play with is the Fifteen Puzzle!

Can you explain why the Fifteen Puzzle is not called a combination puzzle?

CHAPTER 2: THE IVY SKEWB

Now that you know what permutations are, it's time for your first twisty puzzle!  The most famous twisty puzzle is the Rubik's Cube, and we'll get to that later.  For now, we're going to learn how to solve the Ivy Skewb.  It's very fun to twist!

When you get a new twisty puzzle, one of the first questions you have to ask yourself is, "What twists when you twist it?"  On the Ivy Skewb, if you look carefully, you'll see that every twist rotates a corner of the cube along with 3 ivy petals.  See what an ivy petal is?

The next question you have to ask yourself is, "How many different twists are there?"  It looks like there are 8, but there are actually only 4!  Some of those corners are un-twistable.  Just play around with it for a little bit, and don't be afraid to scramble it!

I'm going to teach you how to solve it very soon, but first you have to learn two more fancy words, okay?  These are words that adults usually have lots of trouble learning, but you're kids, so you'll have no problem.  The first one is "commutator," and the second one is "conjugate."  Can you say them with me?

In math, sometimes we use letters to mean different things.  Some of the math people's favorite letters are X and Y, and some of their other favorite letters are M and N, so I'm going to use those when I tell you what the new words mean.  Ready to learn your new words?  Just repeat after me.

A commutator is: X, Y, X-inverse, Y-inverse.

A conjugate is: N, M, N-inverse.

A commutator has 4 syllables and 4 parts.

A conjugate has 3 syllables and 3 parts!

Right.  What if I asked you to build a commutator with the letters A and B?

    A, B, A-inverse, B-inverse!

What if I asked you to build a conjugate with the letters A and B?

    A, B, A-inverse!

Right!  It doesn't really matter what letters we choose.

(Drakeson pointed out that they don't even have to be letters.)

    Dragon, Snake, Dragon-inverse, Snake-inverse!

(Milli was not far behind.)

    Pumpkin, Dragon, Pumpkin-inverse, Dragon-inverse!

(I told you I was only qualified to teach my own children.)

We'll start by practicing commutators!  What are they?

    X, Y, X-inverse, Y-inverse!

Right.  Guess what?  X and Y are going to be twists!

You pick them for me.

Which one is X?  Which one is Y?

They have to be different.

    (They select any two different twists)

If those are X and Y, can you guess what their inverses are?

Show me X.  And X-inverse.  And Y.  And Y-inverse.

Now that you know what those are, do a commutator!

    X, Y, X-inverse, Y-inverse.

That's the magic key to solving all of our puzzles!

Now that you know the magic key, we're almost ready to solve the Ivy Skewb for the first time.

Do you see how any twist X and any different twist Y only overlap by one ivy leaf?  The overlap is called the "intersection" of X and Y.  When the "intersection" of X and Y is only one piece, like this ivy leaf, then the X and Y commutator rotates three ivy leaves around.  That's called a "3-cycle."  There are lots of 3-cycles in our permutation puzzles!

Let me explain a little more about 3-cycles.  Pretend there's a little tea table with three chairs, and Drakeson, Milli, and Mama are all having a tea party, and we're all sitting in the chairs and holding hands.  We could keep holding hands, stand up, step over to the next chair, and sit back down.  The way we shifted to new seats is called a 3-cycle.

We're going to rotate ivy leaves to different faces of the cube in 3-cycles.

The three leaves that are holding hands in the 3-cycle are:

    1. the leaf in the intersection

    2. the leaf that X brings to the intersection, and

    3. the leaf that Y brings to the intersection.

Make sure you don't bring solved leaves into the intersection!

Otherwise, they'll get unsolved.

The best way to look at 3-cycles on the Ivy Skewb is with 3 ivy leaves that have touching tips.  They'll match up on a corner that we can't twist.

(Demonstrations of 3-cycles).

Not only can we control which ivy leaves go into our 3-cycle by choosing which direction our X's and Y's move, but we can also control the direction of our 3-cycle by choosing the order of whether X or Y goes first.

It's time to scramble your Ivy Skewb!

The first step in solving the Ivy Skewb is to twist the corners around until all 6 square ivy leaf outlines match up.  We're not moving the corners to different places because they can't even move to different places in the Ivy Skewb.  That's pretty easy, right?

Now we have to solve the ivy leaves, and we mostly do that with commutators!  Our magic key!  If you choose an X that matches an ivy leaf with a corner, then after the commutator you will have already solved at least one ivy leaf!  And if you also choose a Y that matches an ivy leaf to the other corner of the intersection face, then after the commutator you will have already solved at least two ivy leaves!  It's SO MUCH FUN!!

CHAPTER 3: THE IVY SKEWB & CONJUGATED COMMUTATORS

But.....

If all your ivy leaves are sitting on one twist, we have to fix that with a conjugated magic key!  Let's call that a skeleton magic key.  Do you know what a skeleton key is?  It's a key that can fit in many locks.  Like in a hotel, a skeleton key might unlock all the doors in the hotel.

    In the whole entire world?

That's not what I said.

Conjugates are sort of like that.

Well, actually, they move all the locks TO our magic key and move the locks back!

Remember, a commutator is: X, Y, X-inverse, Y-inverse.

And a conjugate is: N, M, N-inverse.

What if I took the commutator and used that as the "M" in the conjugate?

To help remember how to make a skeleton magic key, you can remember that "N" is in the word "conjugate" and "M" is in the word "commutator."

Here it is:

    N

    X, Y, X-inverse, Y-inverse

    N-inverse.

Ready to try the skeleton magic key?

We need our N to move one of the ivy leaves off of the twist that has all 3 ivy leaves.

See how N moved one of the ivy leaves off the twist like we wanted?  But it also moved 2 other ivy leaves too?  Even though those 2 don't look solved, once we do our N-inverse later, they'll be solved again.  So we won't worry about them.  It's really important to keep track of which 3 ivy leaves we're trying to cycle.

After N, you find your intersection and do your magic key commutator.

Then N-inverse finishes your skeleton magic key!

    (Big shiny eyes full of excitement)

I love you.

CHAPTER 4: THE DINO SKEWB

That Ivy Skewb is so easy now, isn't it?  It's time for the next level - the Dino Skewb!  The Dino Skewb is a lot like the Ivy Skewb, but it's just a little trickier.  This time, there are 8 corner twists instead of 4, and all the pieces look the same!  Each piece has two colors.  But they don't look like dinos, do they?  We could call them "triangle edges," but let's just call them "pterodactyls" instead.  It's more fun that way.

Want to know why it's called the Dino Skewb?

    Yeah!

Because a long time ago, there were pictures of dinos on the sides.

I'll show you what it looked like then.

Anyway, this time, the first step is to solve one side.  The sides are called faces, but to finish a side "the standard way," you're going to have to learn what a Rubik's Cube looks like.  Ready?  There are four rules.  Pay attention to the last one because that's the hardest to understand.

    1. Red is opposite Orange.

    2. Yellow is opposite White.

    3. Green is opposite Blue.

    4. When White is on the top and Red is on the Right, then Blue is on the Back.

If you know that, you can solve any side.  Want to see?

Scramble your skewb and try solving just one side!

Did you check that when white was on top and red was on the right, then blue was on the back?

If not, then you have to rebuild your side!

You did it?!?

Did you notice what happened when you went from the third pterodactyl to the fourth one?  You needed to use 3 or 4 moves instead of 1, right?  And they weren't just any 3 moves!  They were a conjugate or a commutator!

Let's scramble the skewb again and solve another side.

Pay attention to the conjugate or commutator you need for the 4th pterodactyl this time.

Now that your side is solved, you have to solve 5 more pterodactyls, all snuggling together in a clump, until there are only 3 left.

You can start using commutators any time, but you definitely need a commutator for the last 3 pterodactyls.  Commutators are always your magic key!  Just remember to bring the unsolved pterodactyls into the intersection!  And if your X and Y don't have an intersection, you'll need your skeleton magic key again.

Just like with the Ivy Skewb, any two different but overlapping twists X and Y have an intersection of 1, so our dino commutators will always cycle 3 pterodactyls.  Can you do it?  IT'S SO MUCH FUN!!  Mommy needs kisses now.

CHAPTER 5: STARTING THE PYRAMINX

The Ivy Skewb taught us about commutators and conjugates!  Short commutators are magic keys, and conjugated commutators are skeleton magic keys.

The Dino Skewb taught us how to use short commutators or conjugates in blockbuilding, which is what it's called when you start putting pieces together into a block, and you keep them together while you add more and more solved pieces to your block.  It's like when the pterodactyls are snuggling.  In blockbuilding, it's okay to break up your block if you can fix it again right away.

Kids that can solve the Ivy Skewb and the Dino Skewb are definitely ready for the Pyraminx!!  But I have to warn you, the Pyraminx is even trickier than the Dino Skewb.  Do you know what a pyramid is?  The Pyraminx is the shape of a 3-layer pyramid.

The corners are little one-layer pyramids, but they don't move into new places, so they're like the corner pieces on the Ivy Skewb.  How many corners are there?
    4!
That's right. That's also just like the corner pieces on the Ivy Skewb!
The 2-layer pyramids are what move around. How many pieces do those pyramids have?
    5!
Mm-hm.

What can you tell me about the big piece in the 2-layer pyramid?

    It's the core of the 2-layer pyramid.

And how many colors does it have?

    3!

Just like with the Ivy Skewb and the Dino Skewb, a 2-layer pyramid twist X and another 2-layer pyramid twist Y have a 1-piece intersection, so a Pyraminx commutator will make a 3-cycle.

To solve the Pyraminx, we have to do four steps, but you already know how to do three of them.  You still have your magic key, right?

What's a commutator?
    X, Y, X-inverse, Y-inverse!
Yep!

What's a conjugate?
    N, M, N-inverse!
Mm hm!  And what's a skeleton magic key?

    N, X, Y, X-inverse, Y-inverse, N-inverse!

So smart.

Back to our four steps to solve the Pyraminx.

We wont' scramble it yet!  Just listen.
The first step is to twist the 1-layer pyramid corners so that they match the 2-layer pyramid cores.
    (Milli says the solved tips with the 2-layer cores are "diamonds")

So the first step is to make 4 diamonds.

The second step is to twist the diamonds around until each face has matching diamonds.  That's a lot like the first step in solving the Ivy Skewb!

The third step is to use your magic key a little bit, but it's not as easy as it was before.  With the Pyraminx, it's possible for a piece to be oriented the wrong way!  That didn't happen with the Ivy Skewb or the Dino Skewb!!

The orientation of a puzzle piece is which direction the colors on the sides are pointing.

So right now, which direction is my nose oriented?

    Up!

What about now?

    (Pointing and saying a variety of things)

That's right.  You need to remember the new word oriented, okay?

    K.

Before we go ahead and fully scramble the Pyraminx, let's cycle 3 pterodactyls with a commutator.

How can we re-solve the Pyraminx from here?  Well, we can think about this in two ways.  One way would be to undo everything we just did.  That would be the inverse of the whole commutator.  Let me show you.

Now I'll redo the commutator to talk about the second way.

The second way is to forget about how we created that 3-cycle and try to find a magic key that will put the pterodactyls back.  Any of the 3 unsolved pterodactyls can be considered the intersection of an X and a Y.  In fact, choosing the intersection actually chooses the X and Y!  But only ONE of those choices can solve the Pyraminx.  Let's try them all and see which one works.

    (Cuddly puzzle time with Mommy)

With the Pyraminx, using our magic key without paying attention to any orientations won't necessarily solve everything the way it did in the other skewbs!  We can still use our magic key, but sometimes we might want to use a more powerful, bigger key!

The fourth and final step uses big magic keys, and they're pretty fancy.

But you don't get to know about it until you pass your midterm!
You have to get 10 points to pass, and each question is one point.  If you don't get 10 points, you can retake the test one time every day, and you'll always get hugs for trying.

    1. Explain the words combination, permutation, and orientation.
    2. Tell me, in letters or things, what a commutator is.
    3. Tell me, in letters or things, what a conjugate is.
    4. Tell me, in letters or things, what a skeleton magic key is.
    5. Explain what an intersection is and how it relates to a 3-cycle.

    6. Tell me the four rules for where the Rubik's Cube colors go.
    7. Solve the Ivy Skewb for me.
    8. Solve the Dino Skewb for me.
    9. Show me a conjugate and its inverse on the Pyraminx.
    10. Show me a commutator and its inverse on the Pyraminx.
    Extra Credit: Solve one whole face of the Pyraminx!!

Our first midterm exam required a retake because of question 5, which is the hardest and also most important question.  We had to take my cube collection and play the "how many pieces are in the intersection of rotation X and rotation Y" game.  Once we were able to identify intersections, it was easier to understand that

"A commutator creates a 3-cycle if the intersection of X and Y is exactly 1."

CHAPTER 6: THE PYRAMINX & LONG COMMUTATORS

Hi Sweethearts!
You passed your test!!

    Yay!!!

So far, we've been doing short commutators, but now it's time to learn about long commutators.  Long commutators are the ones where X or Y have more steps, like X1, X2, X3 or Y1, Y2, Y3.

What do you think the inverse of (X1, X2, X3) would be?

    X3-inverse, X2-inverse, X1-inverse!

What do you think the inverse of (Y1, Y2, Y3) would be?

    Y3-inverse, Y2-inverse, Y1-inverse!

Great job.
With long commutators, a lot of the time, the X1, X2, X3 are actually conjugates.

So instead of calling them X1, X2, X3, what could we call them?

    N, M, N-inverse!

Right, but what if I wanted to call them all X's?  Then what?

    X1, X2, X1-inverse?

Yes!  That's pretty tricky, huh?

Here's another tricky question.

What if I asked you what the inverse of X1, X2, X1-inverse is?

Could you figure it out?

    X1, X2-inverse, X1-inverse!

Isn't that interesting?  Look at it.  Did you notice that the inverse of a short conjugate only changes the middle?  This part is very tricky, and you're doing a great job.

Now come in close.  We're about to build our first long commutator, which we'll call our first big magic key.  You're going to want to pay attention, because big magic keys help you solve the Rubik's Cube!

If a short commutator, which is a magic key, looks like this:
    X
    Y
    X-inverse
    Y-inverse

Then a long commutator, which is a big magic key, looks like this:
    X1, X2, X1-inverse
    Y
    X1, X2-inverse, X1-inverse
    Y-inverse

Let's just look at that for a little while.

Ready to move on?

Let's try building a big magic key with a short conjugate in the Y.  That would look like this:
    X
    Y1, Y2, Y1-inverse
    X-inverse
    Y1, Y2-inverse, Y1-inverse

Here's the thing - using the big magic key is step 4 in solving the Pyraminx!

I usually use the big magic keys with the long X parts.

Let me show you.

    (Mommy does a skeleton big magic key to set this up)

First we have to find two unsolved edges that are on one face of the Pyraminx.

Also, one of them has to be oriented correctly to that face.

Here we have two unsolved edges on the yellow face, and we're looking for one of them to have the yellow color on the yellow face.  Let's figure out a name for those pieces.

The one that matches the face can be called the chameleon, because chameleons can camouflage to match what they're next to.  Next, we need to name the other one on the same face.  Maybe something a little bit similar because they're on the same face.  What's another reptile pet?

    A bearded dragon!

Thank you, Drakeson.

Milli, you get to name the third piece.

It's from somewhere else - not the yellow face.

So what's another kind of animal?

    A bird.

Okay!  What kind of bird?

    A parrot.

Thank you.

So we have a chameleon, a bearded dragon, and a parrot.

The chameleon and the bearded dragon start off on the same face.

The chameleon camouflages and matches that face.

And the parrot is flying somewhere else, away from that face.

Got it?  Now we're ready to use our big magic key to make a 3-cycle!

The chameleon will go to the bearded dragon's spot.

The bearded dragon will go to the parrot's spot.

The parrot will come to the chameleon's spot.

Make sure you know that really well before we go on.

First we use a conjugate, X1, X2, X1-inverse, to get the chameleon out of the yellow plane, replace her with the parrot, and bring the parrot down to where the chameleon was.

There's only one way to do that correctly and solve the parrot.

Always think about solving the parrot when you're picking your X1 and X2!!

See how the parrot is now in the yellow plane?

THAT'S THE INTERSECTION OF (X1, X2, X1-inverse) and Y!

It's really really important to understand that.

The next step is Y, and we have to bring the bearded dragon into the intersection.

X conjugate inverse solves the chameleon and also solves the bearded dragon.

Y inverse rotates the yellow plane back into place so that all the diamonds match again.

I know it's complicated, but this is the hardest part of the whole course.

To completely finish solving the Pyraminx, all we have to do is get used to using our big magic keys and skeleton big magic keys!!  Without me telling you anything else, can you tell me what that would be?!?

This is a regular big magic key.

    X1, X2, X1-inverse
    Y
    X1, X2-inverse, X1-inverse
    Y-inverse

Remember how a conjugate is N, M, N-inverse?

A skeleton big magic key would put all of that into the M of a conjugate!

    N

    X1, X2, X1-inverse
    Y
    X1, X2-inverse, X1-inverse
    Y-inverse

    N-inverse

Let's go through some more examples together, but first, write down your big magic keys and skeleton big magic keys!!  We can look at your paper while I solve the Pyraminx a few times!

    (Cuddly puzzle time with Mommy)

CHAPTER 7: THE KILOMINX

The Kilominx is one of my very favorite puzzles.  Instead of a being in the shape of a cube or a pyramid, it's a dodecahedron.  Count the number of sides.  How many sides does a dodecahedron have?

    12!

Yes, and what's the shape of each side?

    A pentagon!!

Good job, Drakeson.

The good news about the Kilominx is that you already know everything you need to know.  First you have to do some blockbuilding, and then you have to use some big magic keys and skeleton big magic keys.  That's all.

What's a big magic key?

    X1, X2, X1-inverse
    Y
    X1, X2-inverse, X1-inverse
    Y-inverse

What's a skeleton big magic key?

    N

    X1, X2, X1-inverse
    Y
    X1, X2-inverse, X1-inverse
    Y-inverse

    N-inverse!

Watch me solve it once.  We'll talk about what we're doing every step of the way, and then you can try.  It's even more fun than ANY of the permutation puzzles we've done so far!!

For our skeleton big magic keys, we might need to write down the names of faces for our N's.

    R - red

    O - orange

    Y - yellow

    L - leafy green

    F - forest green

    S - sky blue

    C - cobalt blue (neither one of them had ever heard of cobalt)

    V - violet

    P - pink

    W - white

    G - grey

    T - tan

Do you know about clockwise and counterclockwise?

    No.

Sigh.

    (Cuddly puzzle time with Mommy)

That's it for this chapter.  Now you just get to play with the Kilominx until it's easy.

The Kilominx has 20 pieces and the Pocket Cube, your next puzzle, has only 8.  But the Pocket Cube is NO JOKE.  In fact, when Daddy first met Mommy, he had a Pocket Cube that neither of us could solve.  Mama was determined to learn how to solve it to show Daddy she was smart and win his heart forever!  It worked!

CHAPTER 8: THE POCKET CUBE

The Pocket Cube is really hard, and that's because every twist rotates half the pieces.  When a puzzle is like that, you have to use your keys and trust them, no matter what you're seeing!  Every magic key on the Pocket Cube makes it look really scrambled for a while.

Look at the Pocket Cube.  Tell me what you see.

D:    It looks like a cube with 8 cubies that is really small.

M:    It looks like there has a bunch of pockets.  And it's super hard to solve.

D:    And it can fit into pockets.

M:    And the curved centers look like pockets.

The first step is to choose a color and solve one layer with blockbuilding.

That's the easy part - you just need conjugates for that.

Next, you have to make sure that there are 3 unsolved corners in the unsolved layer.

Sometimes there are only 2, and that's tricky.

If the 2 are in their spots but oriented incorrectly, you need a new tool!!!

It's another big magic key where there are 6 X's instead of 3!!

What should we call it?

M:    A 6-X magic key!

D:    A 6-edged magic key!

Okay, a 6-edged magic key it is.

It makes 2 corners twist!!!

Let me show you what it looks like.

    (Cuddly puzzle time with Mommy)

Sometimes there will be only 2 but they're not in their correct spots.  When that happens, you have to change it into 3, because our keys make 3-cycles.

You already have two tools to turn the 2 unsolved corners into 3.

The first is rotating the unsolved face, and the second is your big magic key.

Let's see what this looks like.

    (Cuddly puzzle time with Mommy)

Once you have 3 unsolved corners, you just need a big magic key or a skeleton big magic key!

I'm going to have you watch a few solves and I'll explain what's going on the whole time.

You'll tell me if we need big magic keys or  skeleton big magic keys.

For our skeleton big magic keys, we might need to write down the names of faces for our N's.

    U - up

    D - down

    R - right

    L - left

    F - front

    B - back

    (Cuddly puzzle time with Mommy)

Kiss kiss kiss.

CHAPTER 9: THE RUBIK'S CUBE BLOCKBUILDING

It's finally time to start the Rubik's Cube!  I know you want to get started right away.

Out of all the puzzles we've done, the Kilominx was the only one that came with solved centers.  But guess what?  The Rubik's Cube is just like that.  It has 6 solved centers, and they follow the rules about where the colors go that we learned when we were doing the Dino Skewb.  Usually, people think of the centers as one big 6-part center piece.

The other pieces are either edge pieces or corner pieces.

How many colors does an edge piece have?

    2!

Right.  And how many edge pieces are there?

    12!

How many colors does a corner piece have?

    3!

And how many are there?

    8!

Right!

And what do you know how to permute - edge pieces or corner pieces?

(Hint - think about the Pocket Cube!)

    Corner pieces!

That's right!

Because edge pieces are the hard ones, we have to start our blockbuilding with edge pieces!  If we started with corner pieces, then we'd just get stuck when it was time for edges.  Nobody wants that.

Most people talk about starting the edge blockbuilding as "building the cross," which is a solved cross shape of 4 edges and 5 centers.  That's fine, but let's not worry about that.  I want you to build a little cube instead, which might be a little harder.  Mommy is so mean!

M:    Mommy is so mean that we can't even go to shapeshifters!-----------    

D:    I don't think Mommy is even being mean at all.

M:    No, you have to say something bad.

D:    No, I don't have to.

First, instead of 4 edges and 5 centers, I want you to build 2 edges and 3 centers, and then fill in the cube with a corner and another edge.  But look, I'll help you.  Watch.

First, we build the frame with 2 edges and 3 centers using short conjugates or magic keys.  That's like a little part of a cross.

Then we can try to fill in the part that's missing, which is a corner piece and another edge piece.

To do that, we have to figure out which pieces we want.

Then we have to put them together.

Then we have have to put them into our little cube with another conjugate or magic key.

    (Cuddly puzzle time with Mommy)

If that's too hard, there's another way.

We can first put the corner into the cube with a conjugate or magic key.

Then we can pull the corner out and put it next to the edge with a magic key.

Then we can put them both back with a magic key.

None of this will be too hard since you can already solve the Kilominx, and you had to do lots of blockbuilding there.  I'll show you.

    (Cuddly puzzle time with Mommy)

Once we have our little cube, we can solve another leg of the cross and fill it in again.

After that, we'll have a block that's 2x2x3 big!

    (Cuddly puzzle time with Mommy)

Once we have our 2x2x3 block, we can solve the last leg of the cross and fill in one more side.  I know everybody else likes to fill in all the sides, but I don't!  We're going to leave one side unsolved, because that's going to help us with the edges.  That's it for today.  In your last lesson, we'll finish learning how to solve the Rubik's Cube!

It's time for Mama sandwich kisses!

CHAPTER 10: FINISHING THE RUBIK'S CUBE

Let's take a moment to talk about everything you know!

You know the difference between combinations and permutations.

You know what commutators and conjugates are.

You know what intersections and 3-cycles are.

You know what the orientation of a piece means.

You know what blockbuilding is.

You know that your magic key is a commutator.

And that your skeleton magic key is a commutator inside a conjugate.

And that your big magic key is a commutator with a conjugate in one part.

And that your skeleton big magic key is a big magic key inside another conjugate!

And that your 6-edged magic key twists two corners!

And you can solve an Ivy Skewb.

And you can solve a Dino Skewb.

And you can solve a Pyraminx.

And you can solve a Kilominx.

And you can solve a Pocket Cube.

And you can blockbuild a whole bunch of pieces on the Rubik's Cube!

WOW!!

You only have two steps left until you graduate!

Remember how Mommy wanted to leave a path in the first two layers of the Rubik's Cube?  That's a really special path that reorients edges without ever using magic keys.  We could think of it like a magical path, really.  We have to use the magical path to solve the last 5 edges, and this is a really fun puzzle all by itself.  Let's call this part the magical path puzzle.

Our first step in the magical path puzzle is to use conjugates with our magical path to orient at least 3 edges on the unsolved layer.  Let me show you.

    (Cuddly puzzle time with Mommy)

Our second step is to try to make it so that exactly 2 of our 5 edges are solved.  That's because we always move things around in 3-cycles, so we need to leave exactly 3 unsolved.

    (Cuddly puzzle time with Mommy)

Our last step is to use one more conjugate with our magical path to finish all the edges!

    (Cuddly puzzle time with Mommy)

And once the edges are solved, do you know what's left?

    What?

Just 5 corners.  And you know how to do 5 corners already because you learned everything you need to know about corners with the Pocket Cube!  Big magic keys and skeleton big magic keys are all that you need now!

Let's try it together!

    (Cuddly puzzle time with Mommy)

Congratulations sweethearts!!

    (Shiny eyes & hugs)

    (GRADUATION RUBIK'S CUBE PRESENTS!!)

Fifteen Puzzle, Ivy Skewb, Dino Skewb, Pyraminx, Kilominx, Pocket Cube, Rubik's Cube

Materials (QiYi brand, Stickerless)

EPILOGUE

Now that you've graduated, you can play with Mommy's Megaminx!  It's like the Kilominx and the Rubik's Cube together, and it's one of my very favorites.  There are also a lot of shapeshifters that are based on the Rubik's Cube, so we can also play with those too.

If you want to try to read Mama's adult posts when you're older, the first post teaches more about edge commutators, reorienting-corner commutators, and reorienting-edge commutators.  The reorienting commutators are even bigger magic keys, because the X in the commutator has TWO conjugates inside it instead of one!  So instead of 3 X's, there are more like 6.  The second post talks about why some permutation puzzles seem impossible sometimes, and how to fix that.  But like I said, that's for when you're older.

I'm very proud of you.