Square-1 Tutorial Part 2: The Solve

When we left off after our painful encounter with Square-1's hideous scramble notation, I promised that I would tell you about the M move, or rather M2, which is an important component in the Square-1 solution. It's one of those rarely-discussed, hard-to-keep straight minutiae of Rubik's Cube notation. Let's say we're looking at a 3-Cube.

We know what R, L, U, D, F and B moves are. And we've discussed the x, y and z whole-cube rotations, that run (respectively) in the same direction as R, U and F turns. But you can also do "slice" moves on the 3-Cube. Not in the Square-1 sense of the word "slice" that I prefer to call "slash" or, better yet, R2, but in the 4-cube-and-up sense of an inner layer of the cube. Like, on the 3-cube, a middle layer. We didn't discuss it before because there's really no need to use it for solving the 3-cube. But here it is, the M move, which moves the middle layer between L and R ... in the direction of an L turn:

Aaaargh. This is another notational quirk that I hate. Why on earth does an x move rotate the cube in the R direction, but an M move turns the middle slice in the direction of L? Forget about killing Hitler. This is what I would change if I could go back in time – stopping whoever decided to make the direction of x and M (and by extention, other cube rotations and slice moves) hard to keep straight. Luckily, there's only one possible result for an M2 move, given the same starting state:

M stands for "middle," I guess. You're turning the middle layer. Meanwhile, just for the sake of completeness, here's the E move – which stands for "equatorial," as in the belt around the waist of the cube; and thank God, it does go in the same direction as a U move or a y rotation.

And here's an S move – which stands for "standing," as in a wall standing in front of you; and it does go in the same direction as as F or z.

OK, let's start trying to solve this thing. Let's look at the top and bottom:

This isn't exactly the same scramble as the one illustrated in Part 1; I shot these photos a couple days after lensing (and solving) that scramble, and I didn't quite succeed in reproducing my previous scramble. No worries; all the steps will be illustrated regardless.

STEP 1: LINE UP THE EDGES. First, try to put all eight edge pieces together on top. Start by taking note of how many edges are already joined together. Above, we see five edges in a row on top, plus a single edge by itself; the remaining two edges form an "L" at the bottom. It helps to put even numbers of edges together on top and to work toward the last two edges at the bottom either being side-by-side or in an "I" shape, on opposite sides of the layer. Here are some pictures of my attempts to get to such a state.

Note: Prepare for each "slice" (or slash, or R2 move) by putting the pieces you want to swap between top and bottom to the right of the "slash" line, and the pieces you want to keep where they are to the left. And bear in mind that pieces at the front, the right of the slash, will end up at the back after the R2 move; and vice versa. If a move you want proves impossible because a corner piece blocks the slice line, see if swapping a front-to-back move to a back-to-front one may help.

So, after floundering around, as shown above, I eventually arrived at a case where I had exactly four pieces in a row, plus an isolated pair on top, and a pair at bottom:

This position apparently made it possible for me to swap the pair up from the bottom, then correct where the pair now at the bottom came back up to the top, allowing me to turn four edges in a row into six.

Finally, moving all six edges on top to the left of the slash, I was positioned to put the pair at the bottom next to them.

With all the edges at top, the bottom layer is all corners. This means we are finally only a few moves away from a cube-shaped cube, or at least, having squares on top and bottom.

It's kind of ridiculous, how many shapes these sides can take, and consequently, solution guides tend to give names to these shapes. I can totally get why this edgeless bottom layer would remind you of a star. Also, that eight-edges-in-a-row shape kind of looks like the Millennium Falcon. But it's probably more useful just to keep track of how many edges are in a row on top, and other cases like an isolated pair, an L or an I.

STEP 2: RESTORE THE CUBE SHAPE, or at least squares up and down. With eight edges in a row at top and an edgeless "star" at bottom, put four edges to each side of the slash:

Do an R2 move, resulting in four edges in a row on both top and bottom:

Next, put two edges on each side of the slash at both top and bottom:

Another R2 gets this "shield" shape, with pairs of edges alternating with pairs of corners, on both layers.

Now line up the slash line between the edges on top and between corners at bottom:

Another slash creates this chevron shape on both sides, with a pair of edges at one end, a pair of corners at the other, and an "I" case in between.

Line up both layers in parallel, with both pairs of edges at one end of the slash line and both pairs of corners split across the slash at the other end.

One more slash, and you have a cube, give or take whatever the middle layer is doing.

STEP 3: ORIENT CORNERS. That is to say, get all four yellow corners on top; which naturally also puts all four white corners at bottom. And remember that from now on, either the top or the bottom layer must be offset by one edge before very R2 move, if you want to preserve the cube shape. Usually, that means twisting the top layer one edge clockwise, like so:

You may also notice that I put the two adjacent yellow corners already at top to the left of the slash line, and I maneuvered a yellow corner on the bottom to the right of the slash.

After an R2, that bottom yellow corner is on top.

With that unpaired yellow corner to back right on top, I put the remaining yellow corner to the front right at bottom, so when I do another R2, it brings the two yellow corners together.

Then do D R2 to line up that yellow bar to the right of slash and dial it up on top:

STEP 4: ORIENT EDGES. Assuming all four yellow edges didn't follow their corners to the top in Step 3, you're facing one of four cases: there's either one white edge on top, or two in an "I," or two in an "L," or three in a "T," or all four, like a white-on-yellow plus-sign. But it's actually simpler than that. If you have a plus-sign, that's just two I cases. If you have a T, that's an I case plus a singleton. So, start by solving the I cases, if any. Here's an example:

The solution is simple: just an M2 move. But how do you do an M2 on this thing? Well, I'll show you. First, put the two white limbs of the I at front and back (both top and bottom) and with the U layer offset by one edge, do an R2 move:

Then, in one of two exceptions to the rule "always offset the U layer after Step 2," realign U and offset D:

Do another R2:

And remember to offset U and realign D again:

So, once again, the Square-1 version of M2, the equivalent of a double L slice turn on a 3-Cube, is an R2, align top and offset bottom, R2, offset top and realign bottom. In Square-1 notation that's / (-1, -1) / (1, 1). Isn't M2 better? Practice that move a few times to make sure you've got it.

Once any and all I cases are resolved, see if you have a single unsolved edge on top and bottom. If so, put the top edge at right and the bottom edge at back.

Then do the algorithm R2 U' M2 U R2, which will create an "L" case. Step by step, here's that first R2:

U':

M2:

U:

And the concluding R2:

With that L case, point the limbs of the ell toward left and back of the top and toward right and back of the bottom layer:

And then, again, do R2 U' M2 U R2.

If there's a possible case where you have an L case on top and an I case on the bottom, or vice versa, I haven't seen it happen yet. I suppose, if it does, you'll just have to strategize about where to put the limbs of your I and L in order to set up a case to which the above steps apply.

STEP 5: PERMUTE CORNERS. We've been using these words, "permute" and "orient," throughout these tutorials. What's the difference, you ask? Well, when you're orienting the corners, you're making sure all the yellow corners are at the top. When you permute them, you're putting corners of the same color together on the same side. At this point, it isn't important if the edge of the same color is between them. If you have "headlights," that means the same color is looking out of both corners on a given side. If one or more side(s) has headlights on only the top or the bottom (bot not both), put a pair of them facing to the left.

Then do R2 U R2 U' D' R2 D R2.

If you have at least one pair of headlights on both top and bottom, put headlights to the front on both layers and do the same algorithm:

After you've done at least one of these moves, you'll end up with all four corners on both layers where they belong.

STEP 6: PERMUTE EDGES. This algorithm will swap the edges at front and right of top and at front and left of bottom. So, it behooves you to strategize which edges to line up for the swap. Like in this case, swapping the orange and green edges on the bottom layer will solve both of those sides, while you simultaneously solve red and green at top:

Once the desired swaps are lined up, do the algorithm R2 D M2 D' R2:

That can be, and in my example solve actually was, the last step necessary to solve the Square-1. See what happens when I realign the top layer and do a simple D2:

However, you may face other interesting cases, like when the edges are all swapped with the side opposite, as in the bottom layer below, as well as the two unsolved layers on top:

In a case like this, you have to strategize a bit to pair up adjacent edges that, once swapped, will set up another L-swap. Then just solve one side at a time, top and bottom if possible.

Now there's an L swap where the bottom-layer red edge can go home, along with the top-layer green edge.

If you're planning well, you can avoid having another "I swap" as a result. After:

Next, I reached this state where the bottom-layer L swap will send green home, creating two adjacent solved sides (green and red), which leaves only an L swap to complete the bottom. Meanwhile, I did get stuck with an I swap (red vs. orange) on top. Since that meant having to keep L swaps cycling on the lower layer, here are some images of me trying to strategize my way out of this situation.

PARITY. About 50 percent of the time, you will face a situation that can't arise on a 3-Cube, and that requires special steps to correct. There's supposed to be a way to detect parity earlier in the solve, but I'm not smart enough to work out the complicated criteria for whether it's parity or not. So, I wait until the edge permutation step and then it smacks me right in the face. If, after solving all the edges on either the top or the bottom, two unsolved edges remain on the other layer, you're in parity-land. Since this is an easier way to detect parity, and the solution is the same regardless, I see no point in overcomplicating my life; just wait till you've reached edge permutation and deal with parity then.

Here's what to do. First, you know that rule about ofsetting the top layer by one edge? This is the second reason to break it, after the M2 algorithm. So, re-align the top layer:

Then do R2 U D R2. Remember, those U and D moves are full quarter-turns.

This shield shape whould be familiar. This time, put both edges on each side of the shield to one side or the other of slash – say, to the left at front and to the right at back – so that when you do another R2, you end up with four adjacent edges on both sides.

Then align all four edges on the top layer to the back left of slash, and all four edges on the bottom to the front left:

Then do an R2, which is (I believe) the exact turn that extinguishes the parity error.

Then pick up Step 2 from where you split the groups of four edges in half at the front of both sides:

And continue from there:

You might have to redo some corner permutation moves before getting back into edge permutation. Here's where I leave out a bunch of pictures I shot, which merely repeat steps I've already shown you. So, you might have to repeat some steps. Think of it as good practice. Eventually, and more surely than if you were squinting at long algorithms in "(-4, 0) /" notation, you'll get to this:

FINAL CASES. Yessiree, that edge permutation step could solve the Square-1. But don't count on it. You could have a case where, after all the edges are solved, the right half of the middle layer is flipped:

Simply do R2 U2 R2 U2 R2 U2 and that's sorted.

Alternately, your cube could end up looking like the entire middle layer is upside-down. What this actually means is that you solved it with white on top and yellow on the bottom, you ninny.

No worries, the algorithm to flip those sides back where they belong is even simpler: R2 U2 D2 R2.

To bear witness to how greatly this approach to Square-1 has clarified its solution for me, I've gone from undergoing multi-day ordeals, filled with loud swearing, to cheerfully managing it in a reasonable number of minutes, multiple times in an afternoon. So, my satisfaction with this puzzle has gone up steeply. No longer do I call it "that bastard." The fact that there's still more to it than most puzzles I do, merely adds to its appeal as a mentally diverting pastime. And despite the annoying limitations of those bandaging effects, it is really starting to become a very comforting cube to play with, with just the right level of challenge – and my particular model feels great in the hands and moves very smoothly. It's such a winner, in fact, that I've gone and ordered (gulp!) the Square-2.