CTO Tutorial

As I discussed in my FTO tutorial, besides the face-turning octahedron, 3D puzzledom also features a corner-turning octahedron (CTO to friends), and it's quite a unique and engaging puzzle. Here it is, front and center, with the three-layer FTO and the Diamond Skewb standing by for support.

Clearly, again, it's an example of that eight-sided Platonic solid, made up of identical, equilateral triangles. But it turns quite differently from the FTO. So differently, indeed, you shouldn't get used to seeing it lying on its side like that. Believe it or not, this is the puzzle you're going to look at nose-to-nose, with a vertex at top, bottom, left, right, front and (though you can't see it) back. Like so:

I don't have any information about the history of this puzzle. I looked for it online, and I came up with articles about different variations on corner-turning octahedra. For example, this page depicts three puzzles that are quite different from the one I got hold of: one looks just like the FTO but turns around the vertices; one is cut more like my CTO but with bigger centers; and one has had its vertices lopped off. The model I own is marketed at Speed Cube Shop as a 4x4 octahedron, making no specific mention of its corner-turning properties. The Cubicle even incorrectly states that it's a face-turning puzzle. But the notion that it's a four-layer puzzle is well founded. See:

Also, I can't tell you anything about speed records or statistics about how many ways you can scramble this puzzle. It isn't a sanctioned event; in fact, I can't even find a scrambler for it. To scramble it, I typically just go to a 4x4x4 cube scrambler and apply it as best I can. It's frustrating to see how little accurate information can be found lying on the ground, in cyberspace; and what little there is, is made hard to find by quirks of terminology.

Each side has three corners, three edges and one center, and each side moves independently of all others, so there's no fixed order in which the colors are arranged. You can turn half the puzzle at a time, since it's cut right through the edges between sides; you can also do "slice" moves, just moving a middle layer, or a regular move, twisting one of the vertices. Unlike on the FTO, any color can get mixed up with any other color in seemingly endless combinations.

Here's an R move, twisting clockwise; of course the counterclockwise twist (viewed as if facing the right-hand vertex directly) is R' (aarrgh-prime).

With similar caveats about "prime" moves, here are examples of L, U, D, F and B moves respectively:

If you recall that red was originially side-by-side with white, this unfortunate picture is meant to illustrate an Lw (ell-wide) move, twisting the whole left half of the puzzle in one go. You'll have to imagine all the other possible wide moves.

And here is an example of a slice move - r to be precise (we'll use lowercase letters for them), twisting the inner layer on the right side only.

...Although in reality, I always do a wide move and then twist the outer layer back, because (it must be said) this puzzle has a marked tendency to lock up, often making it tricky to turn. At least, after being oiled, it turns smoothly, locking issues aside; you just have to make sure all the layers are straight before you twist. And once again, use your imagination to illustrate the other possible slice moves.

For this tutorial, I'm going to adopt a different approach. Instead of scrambling the puzzle and doing a demonstration solve, I'm going to start from a fresh, unscrambled CTO and demonstrate the three algorithms you're going to need and how they work. Basically, STEP ONE is to bring corners of matching colors onto all eight sides; STEP TWO is to put the correct-colored centers on each side; and STEP THREE wraps it up by bringing the edges home. You can actually do all this by applying just three algorithms. It sounds easy, but it isn't. But this tutorial is all about understanding the algorithms, so they can become the easy part and you can devote your brainpower to the bigger challenge: strategy.

All right, so here's a starting state for our example of the corner-swap algorithm.

The forumla is F r F' l' F r' F' l. Mnemonic: "Right, slice up, left, slice up, right, slice down, left, slice down." Behold:

Each time you do this pattern, it will cycle exactly three corners counterclockwise: (1) the lower right corner of the upper left side; (2) the bottom corner of the lower left side; and (3) the bottom corner of the lower right side. Don't lose your place, no matter how stubbornly the pieces lock horns, because it looks like you're just randomly scrambling the puzzle until it suddenly comes together. To get those corners back where they really belong, just repeat the same algorithm two more times.

By the way, there's a "mirror image" version of this algorithm, which I frankly never use, but beware: it rotates the two bottom corners with the lower left corner of the upper right side – clockwise. That formula is F' l' F r F' l F r' (so, "left, slice up, right, slice up, left, slice down, right, slice down"). But I have to emphasize that while it does a mirror image of the prime algorithm, it doesn't not reverse it. I leave it up to you whether to commit this second algorithm to memory or not.

Moving on to the center-swap algorithm, here's our starting state again, followed by the formula F R F' L' F R' F' L, or "right, up, left, up, right, down, left, down" – all vertex moves; no slices.

Notice how this algorithm causes a counterlockwise cycle of the centers on the upper right, upper left and lower left sides; so, not the same sides affected by the corner swap. Again, to undo what I did to get this result, I had to repeat the same algorithm two more times. The mirror-image algorithm, which rotates both upper sides and the lower right side clockwise, does not reverse the above pattern. But, if you want to know it, it's F' L' F R F' L F R' ("left-up-right-up-left-down-right-down").

Two observations are due right here. First, those mnemonics are really helpful; more so than trying to memorize the precise notation for these patterns. If you remember that the corners require slice moves and the centers require vertex moves, you're golden. And of course the third algorithm, coming in a moment, is weirdly a blend of the two. But before we get to that, let's talk about strategy.

The CTO algorithms seem deceptively simple, easy to memorize, virtually foolproof. But actually, they require you to think hard about which corners, centers, or edges you want to cycle, which colors you want to swap. You have to think about how to build a side with two, then three corners of the same color. You have to figure out how to jigger the three sides you want in your cycle into the correct position, with the correct edges or corners in play (which sometimes takes quite a bit of jockeying). You may have to deliberately move a corner or an edge out of place, or cycle two of the same color, in order not to "over-solve" so you will still have a three-cycle to finish with. Finding yourself down to two edges or corners that need to be swapped, when you have to cycle three pieces, isn't the end of the world; you just have to pitch a dummy piece into the mix, unsolve something and work your way back to having exactly three edges or corners to cycle.

The crux of the problem is this middle step, cycling the centers. If at any point you realize that two sides' centers are swapped with each other, you're screwed. At least, that's been my experience. You can try to mix up already-solved centers, rotate centers through different combinations, whatever. But you'll always come back to two sides whose centers are swapped, when you need a three cycle to complete this step. I have struggled with this sometimes for hours, days. I've left the puzzle unsolved for months, in fact, while trying to run down a possible solution to this parity issue. But I haven't come up with anything except to rescramble the whole puzzle and start over. Sometimes I do that and run into the same parity. It's happened to me as many as three or four times in a row. You can imagine the shade of blue my vocalizations turn the air as I scramble and start over, and over, and over. But eventually, somehow – best case scenario, on the first re-scramble – this breaks me out of the vicious cycle of always ending up with two centers that I can't swap without ending up with two centers that I can't swap, etc. etc.

So my appeal is, don't go insane or give up if no strategy under heaven can get you through Step 2. Just rescramble the puzzle and start over. Or, alternatively, if you know a way to resolve this parity without undoing all your good work so far, describe it in a comment below. Seriously. Please.

And so, I think you'll understand me when I say that it's at this two-thirds point, when I've solved all the centers, that I tend to experience a rush of joy and triumph, as if I've clinched the whole puzzle. Because, really, if I manage that much, the rest is just matching up edges. And that, from the starting state below, involves the pattern F r F' L' F r' F' L, or "right-slice up-left-up-right-slice down-left-down." Let me emphasize: slice moves on the right, vertex moves on the left.

As the final result shows, this pattern results in a counterclockwise cycle of the bottom edge of both top sides and the left edge of the lower left side. To undo it, again, I had to repeat the same algorithm twice. The mirror cycle, which does a clockwise cycle of the same two top-side edges and the right edge of the lower right side, is F' l' F R F' l F R', or "left, slice up, right, up, left, slice down, right, down" – slice moves on the left, vertex moves on the right. Can you use these mirror algorithms to reverse the prime algorithms? Sure, but first you have to rearrange the sides to ensure the correct edges are in play. And frankly, I think it's easier just to repeat the prime algorithm two more times.

That's it. That's all I'm going to show you. I could illustrate an actual solve for you, but it would take a lot more moves (and photos) than the above, and I'm not sure how helpful the illustrations would be, with all the other mixed-up colors obscuring what's actually going on. Just trust me (or rather, don't trust me; try it and see), this three-algorithm method, combined with some strategy and a good deal of twisting to get the desired pieces lined up for each three-cycle, will slowly but surely bring order to the chaos of a scrambled CTO. Except when it doesn't, due to that one parity case that, to my present knowledge, can only be escaped by starting over. Once again, if you know better, please tell me.