5-Cube, Revisited

Here's the current state of my Rubik's cube-type puzzle collection, most of which are in the rotation of puzzles that I regularly solve. Some, however, I only touch once in a while, and a couple of them still intimidate me enough that I'm going to let them hang for a bit before I attempt to learn my way around them. But with only a couple of exceptions, which I shall note in a moment, I only have one of each kind of puzzle and I mean to keep it that way. I'm not in this to have a huge collection of all the different 3x3s ever mass-produced, or what have you.

In the foreground, in starring position if you will, is the 5-cube, the main subject of today's update. However, joing us on the patio for just a little sunlight are all its friends. There are five octahedral puzzles at the left. From front to back, they include two Face Turning Octahedra (FTOs), one of which is stickerless and moves very smoothly, the other a stickered model whose movement is very stiff and apt to lock up, and whose stickers started peeling off; I actually peeled off all the stickers on the blue side and tried painting those pieces, with somewhat unsatisfactory results. Third from the front is the 3x3 Edge Turning Octahedron (ETO), which I haven't learned to solve. Don't look at me like that. I just got it last week, and the minute I laid hands on it, some of the tiniest stickers started peeling off. Also, it's even harder to turn than the stickered FTO. In fourth position is the Diamond Skewb, sort of a 2x2 FTO but super-easy to solve, once you have the knack of it. Like the other skewb products on this table, it's designed so that each turn rotates half of the puzzle at a time. Finally, in the back left corner, is the 4-layer Corner Turning Octahedron (CTO), which is also a bastard to turn and, at times, a bitch to solve, but I've learned a way to do it and I'm working on improving my technique. It's actually not so much that it's hard to figure out how to solve it as the three very simple algorithms require a certain amount of strategic thinking that, in one particular case, catches me in a vicious cycle that I can only escape by scrambling it and starting over. I intend to explain my strategies for solving the FTO and CTO in the near future, but not today; it's hot on that patio and I don't fancy shooting more pictures out there.

In the third row in from the left are three dodecahedra: from front to back, the Megaminx, the Skewb Ultimate and the Gigaminx. The Megaminx is a gloriously fun puzzle to do, with 12 pentagonal faces that ... well, I've already covered it on this blog, haven't I? It was the first direction in which I branched out from cube-shaped puzzles, and I still find it most diverting. The Skewb Ultimate, whose moves (once again) rotate half of the puzzle at a time, is one that I've learned how to solve but haven't kept in practice, and now I'm afraid I'll have to learn it all over again if I'm to make any progress on it. The Gigaminx is literally just the Megaminx with an extra layer of moving pieces added all the way around. I haven't started learning to solve it yet but I'm aware that there is also a Kilominx (scaling the concept down to a single layer of corner pieces with no center), with scaled-up versions called a Master Kilominx (with one additional layer all the way around), an Elite Kilominx (with two such added layers) and an "8x8" Kilominx (with three), while the Megaminx-Gigaminx pattern holds with additional layers producing the Teraminx, Petaminx, Examinx, Zettaminx, Yottaminx, Quettaminx and Minx of Madness, which corresponds to a 21x21x21 cube. I'm in no hurry to size up. I still haven't learned my way around the Gigaminx, after all.

The cluster of cube-shaped puzzles at the center of the collection include (at front) the 2-cube, with a smooth-moving 3-cube behind it and a Skewb on top of that, whose funky diagonal turns I'm still learning. I mean, I can solve it, but I still need to build confidence with it. Next to that stack is the stiffer-moving Rubik's-brand 3-cube that started it all, for me, on top of my beloved 4-cube. Behind them are the 6-cube on top of the 7-cube, with which I've become pretty comfortable. The big honker next to them is Evgeniy's Icosahedron, a cheap-ass 20-sided puzzle that I'm pretty much going to display on a shelf from here on. I really wanted an icosahedral puzzle, at least to complete my collection of Platonic solids, but when it arrived last week (in the same shipment as the ETO) I discovered that it's the worst turning piece of junk I own. Dreadful movement, with flimsy plastic pieces and stickers that I'm afraid to manhandle too much. Ere I learn how to solve a corner-turning icosahedron, I'm going to need a better model.

Next to that, in front, is the three-layer pentahedron, with the 2x2x4 Rubik's Tower behind it. They're the only non-Platonic solid puzzles in my collection so far. I've just gotten my hands on the Tower, so I'll start learning it soon. The 3x5 dealie is perhaps the most comfortable puzzle in this collection to hold in the hand; I haven't played with it very much, but I should. At the far right are my three tetrahedral puzzles. At front is the Pyramorphix, a.k.a. Rubik's Pyramid, with the 3-layer Pyraminx behind it and the 4-layer Master Pyraminx, a.k.a. Tetraminx, behind it. I've had lots of fun with, and become pretty good at, the latter two. I was reluctant to get the Pyramorphix because I thought it would only twist at the tips, making it not much of a puzzle, but it turns out to be a shape-changing gadget that turns completely differently from the two pyraminxes. I still have a lot of work to do on it, but I've solved it a few times.

And so, as promised way back here, I have an update about solving the Last Two Edges (L2E) on the 5-cube. I had mentioned that the L2E algorithm for the 4-cube – put problem edges on the front at left and right, then do Uw' R U R' F R' F' R Uw – isn't worth a damn on the 5-cube. I even posted a link to a two-page guide to 5-cube L2E algorithms tailored to every conceivable case. But I really hoped and prayed that I could come up with a solution that I'd be able to carry in my brain without having to memorize all those ridiculous algorithms. You may remember there was also a two-page guide to Last Two Centers algorithms, which my 6- and 7-cube tutorials showed you actually don't need because there's an easy-to-learn, intuitive procedure for solving every possible L2C case, and that also applies to the 5-cube. This gave me hope that I could finally put this L2E problem to rest. That and the fact that I learned that the 4-cube's L2E algorithm actually does work on the 6- and 7-cubes, if you set up the move correctly. So, as I demonstrate how this works on the 5-cube, bear in mind that this procedure can be extended to 6-, 7- and higher cubes.

First, to minimize distraction, I'll show this process on a minimally scrambled cube. Basically, I did the L2E algorithm on the solved cube, and the extent to which it messed up the cube (beyond the edges we're concerned with) will show you that this move does more than just swap a few pieces between the last two edges. There's a reason you do this before you start solving the cube like a 3x3x3; it shuffles a few other things around as well. But it also rotates three crucial pieces between the two edges in question. So, here's the left edge of the two, at front:

And here's the right one:

You'll see that the green and red edge at the top of the stack on the left needs to move over to somewhere on the right edge, where there's a green-and-orange edge piece that needs to move to the left. Well, a very carefully applied version of the 4-cube's L2E algorithm will actually rotate three of these edge pieces clockwise: The top left, top right and bottom right pieces. Which will put all three exactly where you want them, correctly oriented too (so you won't have to repair the edge-parity case that seems to be developing at the right). Here's how.

First, be sure to position the two edges so that the three pieces needing to be rotated are in the correct places, with two matching pieces opposite each other in one of the top layers. This is super-important. If the top left and top right edge pieces don't both belong on the right, you're not going to end up with the desired result. You may have to flip one of the edges to set this up. There's actually an edge-flip algorithm for the right-front edge that's the same as the L2E only without the Uw' and Uw moves that bookend it, but I'm so over that algorithm. All you have to do is dial one of the edges onto the top layer, twist it around 180 degrees and bring it back down on the other side of the other edge. Think, McFly! I mean, "work smarter, not harder." Second, do that initial Uw' move OR just a slice move (u') on the layer where those two matching edge pieces are facing each other. On the 5-cube that'll simply be the U-slice layer, but on the 6- and 7-cubes it could be one or both of two different layers above the equator of the cube; so, twist advisedly.

Then, complete the L2E algorithm with the usual R U R' F R' F' R, and reverse the initial U wide or slice move. The result, if you've done it right:

The other algorithm, and I think (fingers crossed) maybe the only other one you'll need from that two-page guide to 5-cube L2E cases, is where the piece in question is right in the center, which can happen on either the 5- or the 7-cube or, I suppose, any cube with an odd number of layers going up to infinity. Here's what the case looks like with the two problem edges to the left and right of front:

Well, the algorithm I memorized for this calls for those two edges to be at the front and back of the top layer. I suppose this can be adapted to the orientation used for the previous algorithm, but I don't feel like doing it so, pfbt. I mean, rotate the cube to put the two edges on top at front and back.

Then do an Rw2 move (for 7-cubes and up, that means turning all the layers to the right of center), then F2 U2 Rw2 (with the same caveat), then U2 F2 Rw2 (ditto). Result:

For my final update, I'd like to acquaint you with a new (to me) puzzle scrambler that I've bookmarked on my phone, and that I find much more helpful than the one to which I previously linked. Look at this beautiful graphic interface:

It's even more helpful than it looks. For one thing, not only does it show an animation of all the steps to achieve the randomly-generated scramble, but it also shows you what the state of the puzzle should look like at every step, if you click or tap on it. Also, it provides scrambles for some puzzles that my previous scrambler of choice didn't. Here's the menu, which you get to by touching the silhouette of the puzzle above the steps.

One caveat: the Megaminx scramble squeezes that icon off the screen, so you have to reload the site to get another scramble. So, top row from left, you have scramblers for the 3-cube, 2-cube, 4-cube, 5-cube and 6-cube; second row, 7-cube, a 3-cube blindfolded competition, 3-cube fewest moves competition, 3-cube one-handed competition, and the Rubik's Clock puzzle; third row, Megaminx, Pyraminx, Skewb, Square One (a shape-changer I haven't adopted yet), and 4-cube blindfolded; fourth row, 5-cube blindfolded, FTO (the reason I bookmarked this scrambler), Master Pyraminx, Kilominx and something called the Redi Cube, a corner-turning cube I knew nothing about until I reached this sentence. I can already see where my next few shopping trips to the Speed Cube Shop are going to take me. (Another toy I'm drooling over is the Master FTO, which – you guessed it – adds a layer all the way around 3-layer FTO.) Yeah, they saw me coming. And to think this all started when a YouTube video about how to solve a Rubik's Cube caught my eye. Hoo doggy!