Robbie's 4-Cube Tutorial

This will be one of the longer entries in this group of cube puzzle tutorials. One reason is that things start to happen when you add an additional layer of cubies beyond the original 3x3x3 cube. Some additional notation comes into play. Also, the solution starts with a whole series of steps that weren't part of the 3-cube solution—then settles into the eight-step solution for the 3-cube, only with a couple of added wrinkles. Parity cases can arise that you never see with the 3-cube, requiring (in one case) a rather long and complicated algorithm to correct the error; and one or both of them hits about half the time. So get practicing on these new algorithms! The sooner mastered, the more fun cubes 4x4x4 and above will be. Believe me; my favorite, go-to, comfort cube at present is the 4x4x4. By the way, notice this fine example of a stickerless cube. Stickerless totally rules!

Originally marketed as "Rubik's Revenge," and more recently as the Master Cube, the 4-cube was invented in 1981 by Sebestény Péter. I'm not sure but I think that's a Hungarian name, which means the surname might come first. With 16 cubies per side, the Master has no fixed center, as all four layers on each side can rotate independently. This makes possible some 7.4x10⁴⁵ different arrangements of the puzzle. Whatever you call that number, rounded to two significant digits, it's a 74 followed by 44 zeros. Let that sink in. But that doesn't mean it takes bazillions of years to solve it. No indeed; the current world record for the fastest single solve, as I write this, is 16.79 seconds. I say, seconds.

Here are examples of some the additional terminology, notation and moves besides what I introduced in my 3-cube tutorial. A caveat is in order; every internet explainer, tutorial and puzzle scrambler seems to use their own variants of this notation, and I'm not pretending that my version is identical with what they use for sanctioned competition, etc. In different corners of the internet, "slice" moves and "wide" moves seem to refer to the same thing. But in my internal monologue, as I repeat to myself the moves I'm making while I'm solving these puzzles, a "slice" move is when you move one of the inside layers, like the second layer from the right, all by itself while holding the outside layer next to it, and a "wide" move is when you move both layers together. On this puzzle scrambler, for cube sizes of 4x4x4 and up, letter notation followed by a subscript 2, such as R₂, U₂, etc. could mean either a slice move or a wide move; I'm frankly not sure which. I suspect it's intended to mean a wide move because I always go the slice route instead (I just feel it that way), but when I compare my completed scramble to the picture, it doesn't match. This discrepancy doesn't bother me much because I don't plan on ever entering a sanctioned competition, where you have to get the scramble just so. All I care about is starting from a random position. But we're in the weeds; let's get back to the path.

For the purposes of my tutorial and the algorithms presented from here on, let's call the moves where an inner layer turns by itself "slice" and the moves where two adjacent layers move as one "wide." Now for the notation. Outside that puzzle scrambler, most everybody seems to use either a capital or lowercase dubya (W or w) to symbolize a wide move. So this would be either Rw or Lw:

Fw or Bw:

Uw or Dw:

And don't forget that there can be a prime sign ('), meaning counterclockwise, or a big 2, meaning a half-turn, after any of these. Say Fw as "F wide," Lw' as "L-wide prime," etc.

Meanwhile, the notation I like for the slice moves is whatever lowercase letter relates to the side of the cube whose inside layer you're moving. Again, allowing for prime signs and big 2s, behold r:

and l (that's an ell, not a one):

and f:

and b:

and u:

and d:

Say rw as "R-slice wide," u' as "U-slice prime," etc. Is that everybody? I hope so.

SKIP THIS PARAGRAPH, I IMPLORE YOU. Actually, there is a notation for moving the center layers on all three planes, but I find it so confusing that every time I need to know which one does what, I have to look them up. And I don't use them much anyway. Mostly they're for advanced patterns and a few special-case algorithms that come up but rarely. But if you're looking for them, here they are for the record: M, meaning middle, is the middle layer between R and L and, unmodified by a prime sign, it moves in the direction of an L turn (that's right; I told you this was going to be confusing). E, meaning equatorial, circles the waist of the cube between U and D and turns in the same direction as F. S, meaning standing, is the middle layer between F and B and it turns like F. Before you ask how there can be a middle layer when there is an even number of layers, please consider that you can turn all the inner layers as a group, while holding the outer layers in place. Got it? Anyway, there are a few people in history whose shins, if I could go back in time, I would kick; one of them is the yutz who decided that x and M would go in opposite directions.

There's one more point I want to drive home before you get too deep in trying to solve this puzzle. I mentioned before that, unlike the 3-cube, the 4-cube has no fixed centers. So, it's up to you to make sure you not only solve each side, but you solve them in the correct order going around the cube. Since the solution starts with building 2x2 centers on all six sides of the cube, it's possible, but not disastrous, to get them mixed around. I mean, it's disastrous if you don't notice they're out of alignment, because then you'll never completely solve the cube. But it's not too hard to fix. Relax and look at the next two pictures, which illustrate the way the centers should appear in relation to each other.

As I hinted in the 2-cube tutorial, with white and yellow opposite each other and white to your left, the other four sides in ascending order should spell out B-O-G-R (blue, orange, green, red). If your cube has a non-standard color sceme, make a note of it before you scramble.

So, now let's scramble this sucker.

STEP ONE: Solve the first center. By this I mean, create a 2x2, solid-color square in the center of one side. You're free to make that whatever color you like, but I always start with white. This step and the next few after it are somewhat intuitive; another way of saying, use your common sense. But if you want a helpful tip, put the white center together one 1x2 bar ...

at a time.

STEP TWO: Solve the opposite center. Assuming white was the first center, that means yellow. Keep doing that 1x2 bar thing, which isn't hard when you realize you can rotate each side to put the yellow pieces where you need them to be. The trick is getting both yellow bars on the side opposite to white without busting up the white center you so carefully built. There are a few ways you can do this, depending on the case you find yourself in. First, if one of your yellow bars comes together on a side adjacent to white, put white at the bottom.

Then, twist the side with the yellow bar so it points up and down; move that side up to the top (either a wide or a slice move, your choice), give the top layer a half turn, and twist the same side back down. This'll bring a white bar up from the bottom, then put it back while leaving the yellow bar on top.

Second, when you've built the second yellow bar on one of the sides and you want to put it up on top with the first, line it up under the first one, then do a similar up-twist-down move. This will put the second bar at the top while pushing the first one out of the way and, at the same time, pulling a white bar up from the bottom; then, after the twist, it will bring both bars back into place and complete the yellow side.

There's a possible third case, when in order to build a yellow bar, you have to push a yellow piece off the top layer alongside another yellow piece. When this happens, just twist that side to line up the yellow bar with the slot it needs to fill, and turn that side up to put it in place. This will also restore a white bar that got pushed up on the opposite side of the cube when you pushed that single yellow piece off the top. Either way, the result is a solved yellow side:

STEP THREE: Solve the other four centers. There are algorithms but you shouldn't need them. Just apply the methods used in steps 1 and 2 to line up 1x2 bars of each color and put them together into 2x2 squares. I've seen some of those algorithms with M in them suggested for swapping centers that come together in the wrong order. You don't need them. Just keep applying the principle of letting two bars of the same color chase each other around the cube on adjacent sides, and strategically turning one of them into the opposite lane and putting them side-by-side again where they belong. You'll experience some trial and error as you learn this, but once you've got it down, fun with the 4-cube can properly begin. Here are some pictures from representative moments along the way from putting together one bar ...

to seeing the centers start coming together ...

to all the centers being in place.

STEP FOUR: Edge pairing. Now the 4-cube presents another new wrinkle. Instead of having edge pieces ready to go for you like the 3-cube, the 4-cube expects you to build your own edges by pairing up matching colors in 1x2 bars, all the way around the cube. Like, 12 of them. First you have to find the matching pieces and put them opposite each other on adjacent layers of the cube, which may seem tricky enough when you're starting out. In my early days with the 4-cube, I spent a lot of time doing an "edge-flipping algorithm" so matching edge pieces were in the correct position to be paired. And while it is a handy algorithm to know, I find I don't need it so much for its original, edge-flipping purpose. If you want to know, and you should, that algorithm (with the edge to be flipped at front right) is R U R' F R' F' R. But you'll stop using it to flip edges, for the most part, when you realize that you can accomplish the same thing by dialing the upside-down edge up to the top, giving the top layer a half turn, and bringing the same edge down on the opposite side of the cube. None of these maneuvers mess up anything you've already solved, and it lines up the two matching pieces so you can do what this step is really about. First, do a wide move (Uw or Uw', for example) to bring the matching edge pieces together.

Then, twist that side up or down to put that edge on the top or bottom layer; rotate that layer to put an unsolved edge in its place; twist that side back to where your newly matched edge pair was a moment ago; and undo that wide move to heal the center that you had to break in order to pair up that edge. Strategically, you will sometimes need to plan a move or two ahead, making sure there's an unsolved edge on the top or bottom layer to swap with your next, newly paired edge.

STEP FIVE: Last two edges. Obviously, you can't simply pair up the the last two edge pieces, twist them out of the way, put an unsolved edge in their place and restore your centers, because there aren't any unsolved edges left at this point.

Not to worry, there's a move that solves this every time—for the 4-cube anyway. (Ominous foreshadowing...) So, when you get down to the last two colors that have to be paired to complete all your edges, and you haven't lucked out and solved them by accident when you paired the third-to-last edge, do this: Turn the outer layers so the two unsolved edges are on opposite sides (left and right) of front, and the matching pieces for each pair are on the same level. You read that right; this isn't like those other times where they had to be on opposite layers. And I swear you can do this without the edge flipping algorithm, but you do you. Now do a Uw' turn (i.e. twist the top two layers 1/4 turn counterclockwise).

Then do that edge-flip algorithm you're so glad you memorized even though you don't really need it for that purpose. What is that algorithm again? R U R' F R' F' R. Learn it, live it, love it. Oh, and then undo that wide move with a Uw. Voila! Those last two edges are paired.

STEP SIX: Make like a 3-cube. Only move the outer layers from here on, preserving the 2x2 centers as if they were the fixed center pieces of a 3x3x3. You'll see all the familiar old landmarks, like the Daisy ...

and the white cross ...

The white corners ...

eventually leading to the solved bottom layer (here on its side) ...

and the side edges step ...

The 9 o'clock sign (where I think you can begin to see what I meant about 3 and 6 being maybe more important to look at than 9 and 12) ...

and the yellow cross ...

and I swear I didn't forget a step, but my sample solve allowed me to skip directly from Orient Last Layer (OLL) to Permute Last Layer (PLL) ...

and here's the last move of the last step as it all comes together:

Awesome, huh? Well, I said a ways back that there were going to be some extra wrinkles. Parity problems that don't arise on a 3-cube but that start to manifest on each higher size. The two that kick in at 4x4x4 and above are the OLL and PLL parity errors, part of the reason why I've been trying to acquaint you with those terms as we go along.

OLL Parity is this thing that can happen at the end of the Yellow Cross step (F U R U' R' F', remember), when for some reason, the last yellow edge refuses to flip yellow-side-up, no matter what you do.

No moves you learned for the 3-cube will fix this, because it can't happen on a 3-cube. But for 4 and above, it happens a goodly percentage of the time. I have good news and bad news. The goood news is there's an algorithm for this and it works a treat. In fact, you can solve more than one issue with it (more on that when we get to the 5-cube). The bad news is, it's a bitch. It's going to take serious effort to memorize this and, until you've got the moves well and truly locked in, you're going to mess it up (and with it, the cube) about every second or third time you try it. So, get practicing. The algorithm goes like this: With the problem edge at top and front, do r' U2 - l F2 l' F2 - r2 U2 r U2 r' U2 - F2 r2 F2. Got that? And remember, that lowercase notation stands for slice moves. If you do it right, you'll come out of this maneuver with the yellow cross on top and you can move on the PLL step. But that's when you may encounter ...

PLL Parity, where you've just gotten all the top corners aligned, but the top edges won't rotate into place no matter what you do. You never had this problem on the 3x3 either. Either it's because the top edges are out of order, so you can keep doing the final step but they just spin around and around, or it's because you have an impossible number of completely solved sides. You should only have one side, at most, fully solved before you complete the final step. But on the 4-cube, you sometimes find yourself with two solved sides, and somehow this is incompatible with solving the puzzle ... unless you do this (and, thank God, it's a simple and fun algorithm): Putting one of the unsolved sides at front, do r2 U2 r2 Uw2 r2 u2. You read that right: this algorithm has an outer layer move, a wide move and a slice move, all on the top layer. So much fun!

That's the 4-cube. I may have struggled with it a bit, at first, but that was mainly because that OLL parity algorithm is heck to memorize and execute without a mistake until you've practiced it a bazillion times. This was the first 3D puzzle I acquired and learned to solve after the 3-cube, where it all started. I actually bought the 5-cube at the same time and started working on it almost immediately, as well. But even as both puzzles have become like second nature to me, this is still the one I feel I can relax into. It's more challenging than the 3-cube to a degree that makes me enjoy it more, and easier than the 5-cube to a degree that makes me think of it as my comfort cube. Everything I love about cubing is present in this particular puzzle, and sticky issues that irritate me just a bit don't kick in until the 5x5x5. It has a couple of algorithms you need to learn by rote, but it also has more reasoning-based, intuitive steps than the smaller cubes. It's the sweet spot of cubing. And the specific model I bought also happens to be a nice, smooth-turning, sticklerless speed cube that, even though I'm not a speedster, just feels good in my hands. But whatever. If you've got the 3-cube down, think about leveling up.

In summary:

1. Build white center (critical thinking)
2. Build yellow center (ditto)
3. Build BOGR centers (ditto)
4. Pair edges (ditto)
5. Pair last two edges (Uw' R U R' F R' F' R Uw)
6. Solve like a 3-cube
7. OLL parity: r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2
8. PLL parity: r2 U2 r2 Uw2 r2 u2