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How I Solve It
At last we come to the best part, the patterns. Two caveats:
Now, on to the patterns!
There are two dot patterns, one with four dots and one with six dots.
To make four dots I use the sequence αε2βε2. Note that the sequence and the picture are in different orientations—one's good for execution, the other for display. Also, you can think of the ε2 here as a center monoswap, just as you can think of the U2 in (U2α2)2 as an edge monoswap.
There are various sequences one can use to make six dots, for example βωαε for the dots in the picture, but I don't know any intuitive way to figure out which sequence to use to obtain a particular arrangement of dots. So, here's what I do instead. To get the arrangement in the picture (e.g.), yellow needs to move to enclose red, red needs to move to enclose blue, and blue needs to move to enclose yellow. I fix that chain of three faces in my mind, and then I move the yellow face four steps along the chain.
RL-1 FB-1 UD-1 RL-1
However, I also reorient as I go, so that the yellow face is always in front. Let's look in detail at how that works. As I do the first step RL-1, I also do the reorientation LR-1β. The R and L terms cancel, so the net effect is just β. The reorientation turns the second step FB-1 into DU-1. As I do that, I also do the reorientation UD-1ω, net effect ω. And so on, until in the end it turns out that I'm just executing the original sequence βωαε. But, I'm executing it without knowing it, which is a good trick! Also, even though the effect is the same, somehow the process still feels very different.
It's also important to be able to remove the dots after you've made them. It's not hard to figure out the appropriate sequence of slice moves, but I still find the chain-based process to be faster and easier.
The idea of chains is also useful for classifying patterns. Although I only talked about one chain of three faces, the “six dots” pattern actually has two of them, so I'll say that it has type 33. In the same way, “four dots” has two chains of two faces each, so it has type 22. (The two blank faces are left out.) As we progress, it will become clear that this classification is not arbitrary. Patterns of the same type are constructed with selections from a shared set of operations, and are close together in the sense that it's easy to get from one to another.
From here on out, the name “dots” will mean six dots unless otherwise specified.
Checkerboards of types 2, 22, and 222 are easy to make. My sister and I used to call the last one “Purina”. It can be constructed with the excellent sequence α2γ2ε2.
The checkerboard on the left has type 33. It's relatively hard to make, but I'll give all the necessary hints when I discuss mesons and paths. I rate it the second most annoying pattern to leave on someone else's cube. It looks like Purina, but when the victim tries to use the Purina sequence to remove it, ve gets … another checkerboard! That's the middle checkerboard, of type 6. By the way, I'm not going to present any other patterns of types 2, 222, or 6. I mostly like patterns where the centers get moved around, and centers can't move in any of those ways.
If you apply Purina to dots, or vice versa, you get the pattern on the right, which I call “flowers”. Since it has three colors on each face, it doesn't fit into the classification, but it's still pretty. You can make other kinds of flowers (how many kinds?) by applying dots to other checkerboards, but you have to be careful because sometimes the colors will match and you'll get crosses on some of the faces … or even on all of them, which brings me to the next point.
Here are the two easy crosses, of types 22 and 4.
The first is just a checkerboard plus four dots, while the second is one of the places where the edge swap conjugated by RεB-1 is useful. (Or conjugate (U2α2γ2)2 by RεB-1ωR!)
And here are the two hard crosses.
The first is a checkerboard plus six dots, type 33, but the second is something new. It has three chains of two faces each, but two of the chains run through adjacent faces rather than opposite ones. To distinguish this from type 222, I'll let the letter “A” represent a chain of two adjacent faces and say that the pattern has type 2AA. The tricks that are useful for type 4 are useful here too; just change the sign of B and conjugate by RεB (or RεBωR).
[MT] reported the following names for the hard crosses.
Then there are two kinds of cross, known to the MIT cube-hacking community as the Christman Cross and the Plummer Cross …
The names made their way from there to [CG], and from [CG] to me, so I've known them for as long as I've known the patterns. The Plummer cross is the one of type 33.
Type 2AA is the last of the seven types that occur in my list of patterns, but it's far from the last of all possible types. For example, you can construct simple patterns of types 2, A, 3, and C (three faces in a “C” shape) and combine them in many ways. For another example, here's a little story. Just for fun, I thought of the most implausible type I could and tried to construct an example of it … and surprise, I succeeded! So, for your amusement, here are two patterns of type 5 (subtypes 51 and 52).
On a 4×4×4 cube you can move the centers too and get a pretty pattern that I hereby name “heart”.
Unless you can heat them to a temperature approximately 105 times hotter than the center of the sun and so produce a quark-gluon plasma, quarks occur in nature only in the quark triplets known as baryons (protons, neutrons, etc.) and the quark-antiquark pairs known as mesons (pions, kaons, etc.). Rubik's cube corners don't obey the same laws as quarks, but the law they do obey, that the total twist must be zero mod 3, has amusingly similar consequences. It's impossible to twist a single corner clockwise to get an isolated quark, but we can twist three corners clockwise to make a baryon, or one corner clockwise and one counterclockwise to make a meson. The analogy is due to Solomon Golomb and is discussed at greater length in [MT].
So, that's why [MT] and I call the following patterns the meson and giant meson.
The meson is easy to make—just do an unconjugated double twist—but it's pretty nonetheless. The giant meson is slightly harder, but not as hard as you might think. We can take care of the corners with a double twist, and the centers with dots, and then all that's left is the edges. But, the edges just need to move around in two sets of three, so we can put them into place with two edge cycles. For the first set, we can conjugate by RB to bring the edges to the upper face for a slow cycle, or by an additional R2D-1 to set them up for a fast cycle; and for the second set, we can just reorient and do the same thing again.
Both patterns have type 33, of course.
If we make a giant meson but leave out the double twist, we get rings, a nice pattern of type 33.
It's also possible to make rings of type 2AA, which I call “anti-rings”. The anti-ring pattern was originally discovered by Richard Walker.
The worm and the snake were also discovered by Richard Walker. Again, one has type 33, the other type 2AA.
As with the hard crosses, I learned about the worm and the snake from [MT] via [CG]. Both books give full sequences for the patterns, for example this one for the worm.
[MT] doesn't discuss many patterns and gives only one or two other sequences, but [CG] gives a sequence for most of the patterns it discusses. Now, I understand why the authors did that. If you're writing a book about patterns, you probably have to accept that people are impatient. They're not going to read your book page by page and listen to what you're saying, they're going to flip through it and look at the pretty patterns and want to know how to make them. However, thinking about patterns in terms of full sequences is a terrible idea. Full sequences don't just fail to illuminate what's going on, they actively obscure it.
In short, I don't like full sequences. That's why I said at the start that I wasn't going to write out a sequence for every pattern.
The right approach is to look closely at the structure of the pattern and make use of primitives and conjugation. Where the giant meson had two sets of “inner” edges near the axis of symmetry that needed to move around, the worm has two sets of “outer” edges. We can bring one set to the upper face by conjugating by RL-1F-1, the other by reorienting and then conjugating by RL-1F. And that's all there is to it!
By the way, one thing I glossed over here and in the giant meson discussion is that we have to bring the edges to the upper face in the right orientation. For example, conjugating by RL-1F-1 works, but conjugating by R2L-1 doesn't.
Here are some nice related patterns.
The first, originally discovered by Oliver Pretzel, is just dots plus an outer edge cycle. To make the other two, just add one or two inner edge cycles. I'll call these patterns “vortex 0”, “vortex 1”, and “vortex 2”. They're all of type 33, of course.
Patterns don't have to be complicated to be pretty.
There's not much to say about these patterns, but here are two points. First, the one on the left has type 4, while the one on the right doesn't fit into the classification. Second, with a little practice the sequence x = ωU2 that produces the one on the right can be executed in a single motion. That doesn't matter now, but it will matter in just a second.
8. Other Type 2AA Patterns
Because the sequence x is fast, the sequence xα2x-1α is also fast. It makes the first pattern below, which I'll call “sandwich” because the flipped middle layer makes me imagine that the three layers can come apart. Note that the sequence and the picture are in different orientations.
An edge swap conjugated by RεB will get you the second pattern, which I call “present” because it looks like a present with a ribbon and bow. Merry Christmas! Another swap will get you type 2AA crosses.
On the other hand, a vertical edge swap conjugated by U±1 will get you the third pattern, which I call “basket”. (Or try Dα2U-1 on a blank cube!) And what will another swap get you? The same crosses!
Remember I discussed two corner monoswaps, one that twisted the corners and one that didn't? Well, if you perform a swap using the one that does twist the corners and then flip the edges in between, you get two columns that look like they've been pulled out, turned around, and put back. It's not pretty to have two columns on the same face, but as usual you can conjugate to put them wherever you want. I like to put them on diagonally opposite edges, like so.
There are also two ways to pack four columns into the available space. Note the similarities between the last pattern and the basket!
The only column pattern that fits into the classification is the first one, which has type 2AA.
10. Other Type 22 Patterns
Finally, I'd like to show you an example of an edge-corner swap. If you perform a corner swap using the monoswap that doesn't twist the corners and then give the upper face a quarter turn, you'll be one edge swap away from the state on the left.
That's just an intermediate state, not a pretty pattern, but it's very close to the pattern on the right. You can move back and forth between the two with the self-inverse sequence R-1L-1 U2 RL.
That pattern, I think, is the most annoying one to leave on someone else's cube. It only has colors moved to opposite faces, so it looks like it ought to be easy to get rid of with a few half turns, or maybe a few symmetrical quarter turns like (RLFB)3, but in fact it's completely impossible for those methods to work. Because of what I said before about odd permutations, the pattern can only be removed with an odd number of quarter turns!
o November (2012)
@ January (2013)