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.

FTO Tutorial

Here's another tutorial I have tried, multiple times, to create, only to botch the photography somehow and abort. Well, this time I'm plowing through.

When I started playing with this puzzle, I struggled to keep straight which color was supposed to be where. Worse, I kept pounding my head against procedures (and there are several) taught by online tutorials, often requiring you solve the edge-center hexagons and corner-center doublets on multiple sides, all without breaking stuff you'd already put together. After culling this algorithm from here and that strategy from there, I've hit on a method that doesn't require any of that brain-melting fuss. If you're looking for a manageable solve that won't have you cussing and kicking as you eternally put out one fire and start another, you've come to the right shop; read on.

FTO stands for face-turning octahedron, don't you know. And I have two of them, side by side in the foreground of the picture below.

Actually, I have three; the two shown are 3-layer FTOs, but I also have a 4-layer Master FTO which I left out of the shot because, erm, it's scrambled and I haven't quite learned how to solve it yet. Work in progress.

Also in the shot are my other octahedral puzzles. An octahedron is another one of those Platonic solids, the third order of regular polyhedra above the tetrahedron (made of four equilateral triangles) and the cube (made of six squares), but below the dodecahedron (made of 12 regular pentagons) and of course, the icosahedron (20 triangles). You may know these shapes from the polyhedral dice used in role-play games; although there's also a 10-sided die that isn't one of the Platonic solids. So, yes, the octahedron has eight (8) identical sides, all equilateral triangles, and it can be sliced a number of different ways.

For example, the three toys in the background (above) are an edge-turning octahedron (ETO), a corner-turning octahedron (CTO) and the Skewb Diamond. They're all cut differently and they turn differently. I frankly don't play with the ETO because it's covered in tiny stickers that started peeling off the moment I unboxed it, and I'm afraid to lose them. Also, it turns very stiffly. I may have to wait for a smoother-turning, stickerless version to come out before I tackle that beast. I do play with the CTO on a regular basis, though its pieces have a marked tendency to lock up. It may not look it, but it's actually a four-layer puzzle and it's also a bit maddening, because there's one parity error that I haven't figured a way around other than to scramble the whole thing and start over. Finally, the Skewb Diamond is the type of puzzle that, once you learn the procedure to solve it, is quick and easy, almost to a fault. I don't play with it much, simply because I can only focus on so many toys at one time and it tends to get pushed into the background. I have nothing against it, though. Expect tutorials on it and the CTO, coming soon.

But back to the three-layer FTO. I have two of them and, for reasons I won't go into, their color scheme is different. Notice how the green and white share an edge on the stickered one at the left, but meet at a vertex on the stickerless one at right. I frankly don't play with the stickered one. It moves very stiffly and, again, the stickers want to peel off. I actually peeled off all the stickers on the blue side and attempted to paint it blue, with unsatisfactory results. It lives on the shelf while I enjoy, very much, playing with the smooth-turning, stickerless FTO that I bought as soon as it came on the market. Having a puzzle that moves smoothly makes a huge difference, like between hating and loving it. This late model has been a popular release, leading many cubers to advocate for the FTO being made a sanctioned event in speed cubing competitions. I'm all for it.

Now, having eight sides means this puzzle turns in more ways than a cube. And some steps in the solve that I follow require you to hold the puzzle with an edge at front, while most steps call for a vertex (corner, point, as you will) at front. Viewed point-on, here are the basic moves. R (the clockwise move; R' turns the same face counterclockwise):

BR (i.e. back-right), which also has a "prime" move:

L (remember, "clockwise" is judged as if you're facing that side of the puzzle):

BL (back-left):

U:

D (sorry about the focus):

B (recall, the white side is "up"):

And F:

Observe that each side has three corners (at the verticies, with four colors each), three edges (two colors each) and three centers, each with a single color. The centers are pretty much interchangeable, but all the other pieces need to align with the colors of neighboring sides before you can properly solve the thing.

Also, bear in mind that this is a three-layer puzzle. So you totally have the option to move the middle, or "slice" layer, which in my notation I represent by lowercase letters. Here, for example, is an l' move (lowercase ell prime). An "L-slice-prime" is how I'd say it out loud, because it travels in the L direction, on the inner layer next to L. And I won't bore you with all the other letters. To be honest, the only slice moves I use in my solve are l, l' and r. You can work them out, I hope.

The regular octahedron is a funny shape. The sides relate to each other, and move among each other, in a way unique to all other 3D scrambling toys. If you were to cut it in half, you'd get two pyramids with a square base, which is perhaps why you can't just turn a layer onto any adjacent side. You have to skip a side to get to the next one where the edges settle together. So, each of the eight colors can only mix with three other sides – the ones they touch tip-to-tip, but not edge to edge. So, on this particular octahedron's color scheme (shown in full in the next two photos), colors that can mix with each other are purple, red, yellow and blue in one group; white, green, orange and brown in the other. That's weird, innit.

By the way, here's the color scheme of my stickered FTO. Note, again, different colors are juxtaposed, and instead of brown, there's a gray side. (I believe some cubers persist in calling the brown side of the stickerless cube "gray." Nerds.)

It was actually this puzzle that led me to find and bookmark this puzzle scrambler, because the Ruwix scrambler holds no truck with non-sanctioned puzzles. Phooey to them. So, here's a sample scramble, as shown on screen and in reality.

STEP 1. Put together on one side – no, not a full hexagon of centers and edges, but just the three white edges. And if you can remember that, with purple at the back, red should be to the left and blue to the right, you'll be on your way to getting the colors in the right order all the way around. Here I've managed to get the white-blue and white-purple edges on one side (those two white centers are not important).

Here's where I found that last, white-red edge that I need to slot in. It needs some turning to get into a position from which I can do that. I'm not going to give you a blow-by-blow account of how to do that; figure it out.

Here's that red-white edge again, on the left edge of up; it's but an L move away from where it needs to go.

STEP 2. Dial in two (2) of the white corners, taking care to match the side colors of the adjacent edges. It may take some creative, intuitive twisting and turning to jigger them into position to do this. But here's the white corner with blue and purple sides, brought home.

And here's the red-white-blue corner slotted in. Don't worry about the third white corner for now.

STEP 3. Build the sides that mix with white (again, that's green, brown and orange, in ascending order if you're holding white at the left). And by way, throughout this step you will hold white at the left, and keep twisting the left layer so that any moves that cut across that side will go through the corner you haven't solved yet.

This is terribly important. Most of the foul language you will put out during this solve will be due to forgetting to keep that unsolved tip where it counts, and having to back up and redo the white edges.

I always start with green. It helps to try to remember that the green edges need to be shared with purple, red and yellow in clockwise order; or, with yellow at the right and white at the left, first purple and then red in ascending order. Here's me pushing the green-yellow edge (to the right of front) up onto the top layer to get it out of the way so I can dial in the green-red edge there it needs to go.

Now I have to get that green-yellow edge off the top and back into the slot, to the right of front, where there's an orange-yellow edge. That'll mean dialing the right edge up and out of the way and brining green-yellow back down.

If the edges get mixed up (like green-red and green-purple are swapped), there's an algorithm to fix that, which I didn't shoot footage of, because it didn't come up during my example solve. With the side in question facing up, put the two problem edges at left and back, then (carefully!) execute the "swap two edges" pattern r U' R U Rw' U R U' R'. That's with the point toward you, white at the back left and the unsolved white corner pointing up. Remember, the lower-case r is a middle-layer-only "slice" move parallel to R, and the w stands for a two-layer "wide" move. Before and after doing this, check whether the edges are in the correct places, comparing them against the white layer with the half-yellow edge of your top layer at right; there should be two white-adjacent sides where the edge colors line up with those two top edges. Like, in this picture, how the green-purple and green-red edges (at right) are correctly aligned with the white-purple and white-red edges at left.

With the green side's edges in place, now you just have to sort out the brown and orange edges. Remember that with white at back left, brown should be at front and orange facing up in this picture; so those two sides are way out of order in this picture.

Strategically, the first thing I did was dial the brown-blue edge to the left of front, and the brown-yellow edge to the right of up.

Then I did this R move to put the orange-purple edge on top, pushing brown-yellow onto the back, followed by a U' move to put the brown-red edge in position at right to dial down into the desired slot.

After a similar move to swap the last orange edge at front with the brown-yellow edge at up, my green and yellow sides look thus:

And either by good luck or good planning, my orange edges are correctly aligned as well:

Eagle-eyed readers might notice a continuity break in my series of photos at this point. This was deliberate, because it just so happened that when I finished solving the brown and green edges, I got the final white corner for free. For tutorial purposes, I felt it was important to illustrate STEP 4: Putting the third white corner in place. The situation illustrated below presents the simplest version of this problem, where you need to accomplish two things without messing up anything else you've accomplished so far. First, that white corner piece at the back right corner of the up side needs to move to the front corner; and second, but at the same time, it needs to flip over so the white side faces down. Savvy?

There's an algorithm to do exactly this. Your favorite algorithm, "down-down-up-up!" Alternating right and left, and starting from the right, that's a R' L R L' pattern to be exact. I usually try to set things up so I can use this pattern, as opposed to three other possible algorithms, because it's the one I have the least trouble remembering. And here's the result:

Brilliant, right? Yes, brilliant. But let's step back a pace and consider exactly what this R' L R L' pattern did. To start, it rotated the up side by one U turn. Also, it flipped not one, but two of the up-layer corners upside-down. If you number the corners 1, 2 and 3 clockwise from the back left, the corners that get flipped are 2 and 3. Corner 2 is the one that mattered at the moment. But what if that last white corner isn't coming from up-side corner 2? And/or what if you don't want it to get flipped? Then here are three other algorithms, for the record:

• L R' L' R (down-down-up-up starting from the left): This rotates the top layer counterclockwise and flips corners 1 and 3.
• B' R B R': This rotates "up" clockwise and flips corners 1 and 2.
• R B' R' B: This rotates "up" counterclockwise and flips corners 2 and 3.

Notice that these four algorithms come in pairs that exactly reverse each other; so if you keep your head, you can undo a mistake by simply backing out of it.

STEP 5. Orient the yellow corners. Yellow, of course, will always end up on the opposite side of white; so turn the white side down and yellow up (you'll know it by its edges). Here we see a case where corners 2 and 3 of the yellow side need to be flipped.

How convenient that you know that "down-down-up-up" algorithm by heart – again, R' L R L'. Result:

Drat! The yellow edges just happened to end up between the correct corners when I did that. See how the yellow-brown and yellow-green edges are on the same side as their matching corners? It's good luck for that to happen, but it needn't. So, in order to illustrate STEP 6 – permutating the yellow edges correctly – I've had to mess them up on purpose. Like so:

A nice mnemonic for the pattern that corrects this issue is "up-twist-down-twist-up-twist-down-twist." But there are three important things to note before you go off half-cocked. First, notice that for the yellow-orange edge to go between the two yellow-orange corners, the top-layer edges need to cycle counterclockwise. So all the "twists" in that mnemonic need to go in that direction; i.e. they're all going to be U' moves. If they needed to cycle clockwise, those "twists" would be U moves. All of them. In the direction the cycle needs to go. Second, all of the "ups" and "downs" must be done on the right; so, R and R' moves. Third, and most importantly, this is the step where you hold an edge at front, not a point. Try this algorithm while viewing the FTO point-on and you'll be re-solving a bunch of stuff. Got it? Edge forward, R U' R' U' R U' R' U' (or, if the cycle goes clockwise, replace those you-primes with yous). Result:

STEP 7. Clean up the centers. This is, for me, the most fun part of doing this solve. If you find it tedious, you might want to look into one of those other methods, which I abandoned in an effort to cut back on blasphemy. But here's the situation. It's hard to tell from this camera angle, but let's call the brown side in this picture "up" and the green side "back left."

The algorithm I'm about to lay on you will rotate exactly three centers from right to left, looping around from the back to the top again. I could maybe say "counterclockwise" but I'm not sure it isn't actually clockwise. To be 100% clear, the white center at right (on the brown side) will scoot to the left where the green center is now; that green center will migrate to the green side, where the other white center is; and that white center will come all the way around to where the first white center started. Got it? OK, here's the incantation: l R' L R - l' R' L' R. I put the hyphen in there just to highlight the structure of this pattern, for memory purposes. It's slice-down, down-down-up, slice-up, down-up-up, if that helps.

So let's start by sending that green center on the brown side home to the green side. But we don't want to bring another white piece onto the brown side, when there's a brown center right there. So, before doing the algorithm, twist the green side to put that brown center into the corner from which it will cycle on top.

Of course, this temporarily breaks the sides around the edges of the green side, but the algorithm will still work and, what's even more amazing, it won't change anything except to cycle those three centers.

Now remember to undo that move where you twisted the green side. If you're having trouble remembering which layer you turned before the algorithm (and you can twist either the back-left or the up side, or both), observe which edge has a bar of color (red in this case) and slot it back home. (If you twist both layers before the algorithm, undo those moves in reverse order afterward.)

Here I use the same algorithm to cycle the red center from the yellow side ("back left" for algorithm purposes) to the red side ("up"), while moving one yellow center on the red side one slot to the left, and the other yellow center onto the yellow side. No preliminary twists are necessary to set it up.

Here I push the remaining yellow center from the red to the yellow side, at the expense of bringing a blue center over to red. To start, I twist the yellow wide to bring a blue center into play; to finish, I undo that twist.

Here I want to cycle a blue center from the yellow side to blue, while bringing a yellow center back. This time, I have to do a U move to prepare for the algorithm, then undo it afterward.

It's nice when you can simply swap centers between two sides where each belongs, but you don't always have that option. Here, I make a small sacrifice to exchange the white center on the brown side for a brown center on the orange side, starting with a temporary U' move.

Here's an opportunity to swap a white center from the orange side with an orange center on the white side – starting, again, with a preparatory U' move to be undone at the end.

See how this cycle works? Here's another example, where it allows me to swap a white center on green with a green center on white, no preparation necessary:

Likewise, green-on-orange vs. orange-on-green (we're coming into the home stretch here):

If I told you without showing you, would you believe that I did a BL' move before the center-swap algorithm to trade a blue-on-red center for a red-on-purple one?

Now it's all about swapping two blue-on-purple with two purple-on-blue centers. The "up" (purple) side won't have to twist to put both of those blue centers in play, one after the other; remember how the cycle works. But there will have to be a back-right twist to set up the second cycle. See:

I think this is a super-fun puzzle. The stickerless version with its smooth turning, worthy of a speed cubing competition, has been a hit since it came out. All the cuber YouTubers are talking about it. And like I said, there are multiple methods for solving it. If you want to delve into one of the tricker ones, like where you pretty much have to solve one side at a time, etc., go for it. There are tutorials for them out there. I'm just saying, I watched them, tried to follow their advice and got terminally stuck ... until I sussed out enough of the patterns I've shared here to save myself a lot of headaches. And despite the algorithmic nature of this method, it remains a wonderfully strategic puzzle. You have to make choices, like which to use of the four corner flip/rotate patterns, which centers to set up for the three-cycle and even, maybe, whether to preemptively put the green, brown and orange edges in the correct order or to fall back on that long, chancy edge-swap algorithm. You have moves that, if done correctly, only touch the two or three pieces you're concerned with – a marvel of mathematics, in my books. And of course, there are less clean moves, that mess up one thing to fix another; but you do those early in the solve when that doesn't matter so much. And when you make a mistake and have to go back three or four steps, it's more practice. Which is all good.

I can't wait to tell you about the CTO. It is so wildly different from the FTO, it'll blow your mind. And maybe, when I get to the part about the parity error that I don't know how to solve, you'll be able to chime in with a solution. Till then, keep your octahedra dry!