Hi Twisty Puzzles fans,

Enabler Cube was suggested in a discussion at the Twisty Puzzles Forum. It acts like a Rubik's Cube with blocked layers. Only three layers can turn, i.e. those in which the rounded "enabler" cubie resides. Specialists may call this type of turning "deeper-than-deep cut". This puzzle is sort-of the opposite of Fused Cube, where the three layers of the special corner cubie cannot turn. Like Fused Cube, also Enabler Cube is a doctrinaire twisty puzzle.

This puzzle was inspired by a similar design by Evgeniy Grigoriev with an unlocking edge instead of an unlocking corner, and a comment by Bram Cohen.

Watch the YouTube video.
Buy the puzzle at my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar

Fantastic puzzle Oskar, this looks like a hard puzzle to solve. I honestly have no idea how the blocking/unblocking mechinism works, would you be so kind as to provide some pictures? As for the number of possible states of this puzzle, I am unsure if it has less states than a standard 3x3, although my instinct is no, if there is both a corner 3-swap commutator and a 3-swap edge commutator, you should be able to put this cube into any valid 3x3 position.

Hmm. So it's an order-1 puzzle with stored cuts.

Great puzzle, I'd imagine it to be reasonably hard.

Since the enabler has to move with every move you make, it's a lot like a maze.

Could you consider this to be an unbandaged Bandaged Cube? That puzzle also has the same property (only faces with the single cubie can turn).

Another GREAT puzzle. And this one is probably second on my wish list at the moment. And that is due to this...Being just as big a fan (if not an even bigger fan) of mechanisms as I am of the puzzles themselves, I really want to understand how this mechanism works. I've been studying the pictures on Shapeways (where you can actually see most, if not all of the mechanism) for a while now and I'm still not sure how this works. I do notice the foot on the special corner is different then the others but you do have to rotate the 3D model a bit in order to see it. It's bulkier then the others. The puzzle also appears to have an inner and an outer set of edge pieces and I suspect its these pieces that are doing the blocking and the enabling and maybe why Oskar said the name is a little misleading in the video. I'm guessing the extra bulk of the special corner allows it to push these inner edges and allows the moves but I think I'd almost have to see it in person to exactly understand what is going on here... so I'm very very tempted to get this after the Clockworks 4x4x4. That is another puzzle I want more for the mechanism then I do the puzzle itself as I'm sure I can't solve either of these by myself.

Carl

Well done!

I was actually thinking about this puzzle a while ago when playing with my keychain cube. If you only make turns by pulling the keychain you have the exact same puzzle. It was impossible for me to solve it that way, but it was great fun:)

It's awesome to see this puzzle made in reality!

PS. This is a totally new type of bandaging, right? Could it be made with more corners/edges being bandaged?

If to place a blocking corner in a position BDL or FDL, it is possible to solve this cube by this method:
http://twistypuzzles.com/forum/viewtopic.php?f=8&t=24137&start=0

Looking at my comment now, I'm not sure what I was getting at, but I'm glad you found it helpful!

I'm a big fan of the deeper than deep cut idea, even if you don't see it that way.

I'm super curious about how many states it can reach (super and non-super stickered). I'm not sure what would be more amazing, if it had the same number of states as a fused cube or a 3x3.

/wait for Andreas with amazing GAP skills

I don't think this is a puzzle GAP can handle.

I'm still trying to get a general handle on the size of the number of states. If it seem small enough I might write a program to do God's algorithm for it.

I haven't actually done anything with GAP, but I definitely think it can be done. You have to think about the puzzle slightly different. Oskar thinks of the puzzle as being able to turn any face that has the enabler cubie. Under that scenario, yes, I would have to agree that you probably can't do it with GAP.
Alternatively, thinking of this as a deeper than deep-cut puzzle, you keeping the enabler cubie as a fix holding point that never moves. Compared to the Fused Cube, the enabler cubie is equivalent to the 2x2x2 bandaged block. There's only 3 faces you're allowed to move and you must move the outside face with it's adjacent slice at the same time.

Another analogy that might work better for you is with gb 3.1.31. If I recall correctly, while you were solving the 2nd set of edges (or maybe it was with the 1st set), you used inverse moves where you would make a turn, then rotate the entire cube the opposite direction.

If neither of these analogies helped, I'll make up some pics.

Very creative. I agree this representation can be done in GAP.

I've done some testing and I'm going to go out on a limb here and say I think this puzzle has the same number of states as a Rubik's cube.

I've found enough pure 5-cycles, 7-cycles, 2-2 swaps of the edges that I think composing them should be able to reach all of the permutations of the edges. The same is true of the corners. I've also come up with a good corner-twisting sequence:
[R', F']x35, U, [F', L']x35, U', [R', F']x35

I'm not seeing any obvious new restrictions.

EDIT: Thinking about this as your double-moves, the permutation parity of the corners is attached to the permutation parity of the centers (basically only 12 orientations are reachable for one parity of the corners and the other 12 orientations for the other parity of the corners).

Keeping a corner fixed I think the size calculation should be:

? choose(n,r) = n! / (r! * (n - r)!)
? 7! * (3^7 / 3) * (12! / 2) * (2^12 / 2) * ((choose(6,1) * choose(4,1)) / 2)
% = 21626001637244928000

This is half the Rubik's Cube group size.

EDIT 2: It looks like Jaap has already analyzed these things and beat me to the calculation. See:
http://www.jaapsch.net/puzzles/subgroup.htm
Jaap calls turning an outer face and slice at the same time a "wide move". If you scroll to "3bj" in the wide-move table you'll see:


Where Uw, Rw, Fw are the generators. The DBL entry is the corner that stay fixed. Jaap doesn't explain where / 2 restriction at the end comes from but I think it is the corner parity + center orientation linkage.

There is nothing I can add to Jaaps analysis.

Here you are. Each (red) edge has a small (yellow) locking pieces that is sliding inside. The (green) enabler cubie pushes the locking piece into other (blue) corner cubies, see pictures below.

Oskar

So the "enabler" essentially blocks the movement of a blocker from the opposite layer. Very clever!

Mind you, I would prefer no stickers on the enabler, just to make it a little bit more tricky - as if I could solve it anyway.

The enabler cube is a great work of art and a great puzzle Oskar, what I love about your puzzles is they are totally unique and a great piece of modern art.

Thank you so much for these images. So if you had a puzzle with 8 of the normal (non-enabler) cubies, if I understand things correctly you wouldn't be able to make slice turns. Is that correct? You'd first need to complete a face turn before you could start the opposite face turn.

Also look at this picture:

Why is the connection part (circled in red) of the yellow slider not the same width as the inner most shell layer? Is this to give the long narrow part in the outer shell layer additional flexibility to aid in assembly?

Just curious...
Carl

very interest mechanism! thank you Oskar
Excellent Puzzle

Nice mechanism!

It took me only three times to look at the pic to understand how it works.... (in different times)


Alex.

So, then, how could you make a higher-order version? Say, only the 2 layers on each side containing the enabler could move on a 5x5 version...

Edit: Or, what about a FTO version? I wonder if it's possible. If so, I propose the name "Blocktahedron".

Indeed.No. The yellow pieces slides up and down. This is the largest connection part that fits within the edge piece.

Oskar

Very clever way to have a locking mechanism which doesn't require spring loading. I wonder what else it could apply to.

Oskar, that is a very creative mechanism. As Bram and Jared said, I believe that this concept (and mechanism) could be applied to other puzzles.
Also, I think this is the first time I truly understand how a complex mechanism works. Thank you for this, Oskar.

-Doug

P.S. Non complex mechanisms include Shells, Knucklehead, etc.

Hi Oskar, I always love your puzzles, but they are out of my price range, does Shapeways have any other materials that are cheaper than white strong and flexible? Also, have you ever thought about making puzzles smaller so they are cheaper?

Ok... I see it now. Thanks. And while I'm here I'm curious about something else...Is the mechanism Evgeniy Grigoriev used for his "unlocking edge" known? If its been shown, could somewhere point me to it? Or is it just "similar" in function? To my knowledge it hasn't been revealed yet. And I'm guessing the trick may be even simpler in his as I don't believe he has any 3D printed parts in his Jailbreak Cube. It appears to be a hand modification to a commercially available 3x3x3.LOL!!! I couldn't follow the idea back when you first said it. Not saying there isn't something there... even the solution Oskar came up with, I still didn't fully understand until I'd studied his posted pictures for a good long while.

After seeing this I now wonder if something similar could be used to make a version of Oskar's Treasure Chest which required all 3 layers to be solved. Note with these sliders under the edges you now have a way to get information from the bottom layer to the top/removable layer. I'll have to think about that a bit more...

Carl

I also think it can be done, at least for the corners, however in the first variation i though about it wouldn't be so elegant,

Alex.

Fantastic puzzle Oskar! I am amazed at what a short time it took from my original post regarding deeper-than-origin puzzles to the physical implementation of the two new concepts offered in the post. Incredible! Also, many thanks for the images showing the ingenious mechanism.

There are a few questions left unanswered from the original post though:

First, is there agreement that these puzzles are deeper-than-origin? There are now four unique puzzles in this category: RotoPrism 2, Trapentrix, Bingo Cube, and of course the Enabler Cube.

To answer that question, it would help to have a clear definition of "deeper-than-origin". I'm happy with the definition originally proposed, which is something to the effect of: "For a puzzle to be considered deeper-than-origin turning (or sliced), a segment of the puzzle that includes the origin point must be turned in order for the puzzle to maintain the orientation of its axis system." Does that work, or should it be clarified further?

The last question I have (and perhaps the most interesting) is whether anyone has identified any other potential deeper-than-origin geometries?

There is still debate on the term "deep cut" so I'm not sure you'll get the agreement you are after. I for one understand were the term, deeper-than-origin, comes from and I think it fits nicely so I'm happy to use it. Just use it in a consistent fashion and I think you'll be ok.Yes... maybe... Looking at the Enabler cube as an example... and Jared's post above. But couldn't this exact same concept be applied to a Megaminx? Imagine a Megaminx with an Enabler corner. Wouldn't it also be a deeper-than-origin puzzle? This could probably be applied to the corner of any face turn puzzle. The concept anyways... I think you'd need a new mechanism to make it work as I don't see how to apply Oskar's mechanism to the Megaminx.

Looking closer at Jared's idea of a 5x5x5 version where you can only turn the 2 layers on the side containing the enabler I think you still end up with the exact same puzzle. I.e. this 3x3x3 Enabler Cube... it just now has very big corners and a narrow slice layer. At least that is what I think you get...

Carl

Bram describes how a bandaged 3x3 can behave partially as a deeper than deep cut puzzle here.

I'd mention the octahedron and dodecahedron, but Carl beat me to it. I guess that leaves only the tetrahedron to talk about. Oskar's mech should work there too I think, but I think there's a better solution there. Tetrahedron's aren't my strong suit and really confuse me. 5 minutes ago I swore you could just glue a piece to the core, but now I can't figure out why I thought that.

Oh, a trivial things to do it on would be the floppy cube. Although you could hardly call that a puzzle anymore because it would only have 24 states and because you only have 2 moves, it's state graph would just be a ring.

In general a puzzle can be deeper-than-origin if the axes aren't opposite each other. It is of course possible to take a subset of the axes of a puzzle where they're opposite, as in several of the existing cases. I'm really curious what a deeper than origin version of the jumbleprism (corner turning triangular pyramid) would be like.

Solving this one is interesting. You can solve the corners using any old method for solving a 2x2x2. Once that's done, there's a simple sequence for swapping two pairs of edges: put the enabler in UFR, then to (R2B2L2F2)3F(R2B2L2F2)3F-. Once you optimize out the extra Fs that sequence is a measly 24 slices long.

> corner turning triangular pyramid
Bram, you're talking about this puzzle?

Wheel Pyraminnx (sorry, not published - bad turning)

No, I mean this puzzle - http://www.shapeways.com/model/397623/jumble_prism.html

I don't think that is always true. Look at the face turn tetrahedron. You start with shallow face turns but once you are "deeper-then-origin" you just have a puzzle with shallow corner turns. The symmetry of the turning axes also plays a factor.

Carl