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 Post subject: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 10:36 am 
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http://www.shapeways.com/model/930817/i ... Box-search

This just went up on his page recently. I can't wait until it's printed!

Any clue how to solve it?


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 12:25 pm 
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I did some looking and found these useful algs:

Permutation: 15-move single-swap
[U,R]x7,U (plus a reorienting of the puzzle)

Orientation: 40-move 2-corner twist
[R,U,R,U,F]x4, [L,U,B,U,B]x4

I think these two algs are sufficient to solve the puzzle (maybe inefficiently though). You can use the permutation alg to setup for the orientation alg. It may get tricky because the first move of the next alg cannot be the same as the last move of the previous alg, but it helps that the permutation alg has two choices for which move to start with due to symmetry.

Also...

CCW-turn: 79-moves
[[R,U]x7,R,F, [R,U]x7,R,B]x2, [R,U]x7,R


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 1:13 pm 
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Wow, this absolutely must be mass produced!!
I was wondering if anyone would ever come up with a twisty puzzle that can't be undone the way it was scrambled.

-d


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 1:38 pm 
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I'd put a lot of thought into making a 3x3x3 irreversible cube for quite awhile now. What's been getting me was how to prevent repeated use of the same twist, but I see Oskar's come up with a good elegant solution.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 2:46 pm 
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darryl wrote:
I was wondering if anyone would ever come up with a twisty puzzle that can't be undone the way it was scrambled.
Every quarter turn of Latch Cube requires three quarter turns to undo. So technically, it cannot be undone the way that it was scrambled.

FYI.
Christoph Lohe (by email) wrote:
Btw, I had meanwhile discussed this puzzle with Jaap Scherphuis, and his findings are:
- The number of states is the same like the normal 2x2x2 orbit size: 3,674,160
- Some states require 37 moves to solve! ("God's number")
- The shortest identity has length 14. It means after a first move from the solved state you will need 13 further moves, at least, to go back to the solved state (Jaap did not calculate what exactly this sequence is).
Oskar

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 4:33 pm 
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Oskar, you should definitely talk with Uwe about mass production of this puzzle.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 4:49 pm 
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What I love most about this is the extreme difficulty vs extreme *apparent* simplicity: this will be a lot harder to solve than the 3x3x3, even though it looks a lot easier. It is utterly misleading, and I think that's what makes the best puzzles, and made the 3x3x3 itself so popular. This should definitely be mass produced! :wink:

Another thing I just realized is that all solving algorithms are based on a binary tree: after each clockwise quarter turn about one axis, there are only 2 possible moves, i.e., a clockwise quarter turn about either of the other 2 axes. So every turn is a binary decision leading to another binary decision. I love it!

This is the BEST, most perfect puzzle ever!!!

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Last edited by KelvinS on Thu Feb 21, 2013 5:06 pm, edited 8 times in total.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 4:57 pm 
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This puzzle is a very exciting variant, I love this puzzle! Now the question becomes how well would this type of mechanism translate to a 3x3x3? ( chills flow down the spine if this can happen)


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 5:12 pm 
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I also love how this puzzle could create huge stress/tension - just one accidental bad turn and ... bugger!!! :lol:

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 5:23 pm 
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This is certainly a very interesting puzzle, but a few things would need to happen for it to go into mass production. First a physical prototype needs to be made which works well, which isn't a done deal, and second and more importantly it needs to be something which is super compelling to the general public. My guess is that it simply isn't obvious enough what makes this one interesting when you look at it sitting in a box for people to impulse buy it. Unlike, say, the gear cube, whose concept 'A Rubik's Cube with gears' is both trivial to see and understand and it looks quite interesting.


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 5:34 pm 
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Bram wrote:
My guess is that it simply isn't obvious enough what makes this one interesting when you look at it sitting in a box for people to impulse buy it.
Nothing that a bit of creative marketing with social media couldn't fix! :wink:

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 5:58 pm 
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Very, very nice puzzle as always! :D

Just a thing I thought of though: When solved, can you make all the three turns, or just two of them (Since your last move to solve it should have bandaged one of the turns)?

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 6:04 pm 
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Temeraire wrote:
Just a thing I thought of though: When solved, can you make all the three turns, or just two of them (Since your last move to solve it should have bandaged one of the turns)?

Good point, I guess there must be 3 different solved states with alternative pairs of turns enabled, depending on the last turn made.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 7:13 pm 
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Oskar wrote:
Christoph Lohe (by email) wrote:
Btw, I had meanwhile discussed this puzzle with Jaap Scherphuis, and his findings are:
- The number of states is the same like the normal 2x2x2 orbit size: 3,674,160
- Some states require 37 moves to solve! ("God's number")
- The shortest identity has length 14. It means after a first move from the solved state you will need 13 further moves, at least, to go back to the solved state (Jaap did not calculate what exactly this sequence is).
I'm glad Jaap saved me the time :lol:

So does this puzzle form a group? I think it actually does because even though U is one move, there is still a U' in the group, even though it's composed of many moves. I think all elements of the group have an inverse even though their inverse is not a simple inversion of the move sequence.

Very strange and very cool.

Also, a god's number of of 37 is not surprising since the lower bound is already quite high:
log((8! * 3^7) / 24) / log(2) = 21.8

With so many restrictions it isn't surprising that it takes 37 moves to reach everything.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 7:19 pm 
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Why not just use the same idea that keeps V6's inner layers aligned to convert this to a 3x3's core: you wouldn't even have to modify the edge/corner pieces. However, after one U turn, it would block both the top and bottom faces... I'm terrible at explaining this kind of thing. :(

Anyway, awesome puzzle. This is at the top of my list of puzzles I want to see mass-produced.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 10:20 pm 
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bmenrigh wrote:
Oskar wrote:
Christoph Lohe (by email) wrote:
Btw, I had meanwhile discussed this puzzle with Jaap Scherphuis, and his findings are:
- The number of states is the same like the normal 2x2x2 orbit size: 3,674,160
- Some states require 37 moves to solve! ("God's number")
- The shortest identity has length 14. It means after a first move from the solved state you will need 13 further moves, at least, to go back to the solved state (Jaap did not calculate what exactly this sequence is).
I'm glad Jaap saved me the time :lol:

So does this puzzle form a group? I think it actually does because even though U is one move, there is still a U' in the group, even though it's composed of many moves. I think all elements of the group have an inverse even though their inverse is not a simple inversion of the move sequence.

Very strange and very cool.

Also, a god's number of of 37 is not surprising since the lower bound is already quite high:
log((8! * 3^7) / 24) / log(2) = 21.8

With so many restrictions it isn't surprising that it takes 37 moves to reach everything.


Sure it's a group, it's the same group as the normal 2x2x2 cube. It's just presented in a rather inaccessible way. If all elements (possible positions of the cube) didn't have inverses, then this would be useless as a puzzle -- it would be possible to scramble the cube into a position which could not be solved without disassembling/restickering the damn thing...

On the normal 2x2x2, we imagine the group as generated by U, D, R, L, F, D (or any 3 non-coplanar turns, if you want a minimal set of generators, like <U,R,F>). Any element of the cube group can be represented non-uniquely as a word in those letters in the obvious way -- the word UURF represents the position on the cube you'd get by performing U twice, then R, then F. But the moves you physically make on the puzzle are not the elements of the underlying group; the elements of the group are the possible orientation/permutations of the 8 cubies. There are lots of other sequences of moves that result in that same position, like UURFLLLL. That tells us that the presentation of this group is something like:
<U, D, R, L, F, B | U4, R4, F4, (LR)^4, ...>
Where the relations on the right are some sufficiently long list of identities.

On this puzzle, clearly the cube can be in exactly all of the same states (and no more). What's different is that the state which corresponds to UURF can no longer be reached by turning the U face twice, then the R face, then the F face backwards, since those moves are blocked. But there's infinitely many other finite sequences of moves not involving double turns that result in that same position, UURF. Doing any one of those one the normal 2x2x2 would clearly result in the state UURF as well. So in a purely algebraic sense, this puzzle is still "generated" by <U,R,F>, but not all words in that alphabet of symbols correspond exactly with the sequence of turns it seems to represent. The presentation of this group looks radically different: the generators are the same, but the relations (identities) are incredibly unintuitive -- without actually playing with this thing I can't think of any other than silly things like (URF)^4. But the presentation of a group is not the group itself; the two are still isomorphic in this case.


I doubt it's possible to build a twisty puzzle whose underlying structure is a monoid that's not a group. (There has to be a solved state (identity element), any scramble you make can be broken up into a sequence of two shorter scrambles without affecting the outcome (associativity), and performing any two scrambles must result in a legal position (closure), which is exactly the conditions of being a monoid -- that much is necessary just from what we mean by "twisty puzzle". Monoids with inverses for every element are groups.) As I mentioned above, you probably wouldn't even want such a puzzle, since the joy of twisty puzzles is knowing there's a solution to the arbitrary scramble you're given and looking for clever ways of finding a finite sequence of moves that works. If there's a real chance that solving a given scramble isn't possible, what's the point?


Anyway, Oskar, this is an amazing puzzle. I sincerely hope it gets mass-produced at some point; I would love to own one (and don't have the disposable income to indulge in shapeways orders). Bravo!


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Thu Feb 21, 2013 10:30 pm 
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Thanks Iranon. I'd never heard of a monoid. It seems the only difference between a monoid and a group is that all elements of a group have an inverse.

The reason I had any skepticism that this puzzle formed a group is that the Rainbow Nautilus doesn't form a group due to the bandaging (it does have the invertability property though). I was going to suggest that there may be two states that are farther away from each other than 37 moves but as I thought it through I thought the puzzle actually probably does form a group even though moves aren't invertable and if the puzzle's states form a group than 37 moves should be the farthest between any two states.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Fri Feb 22, 2013 2:26 am 
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If the permutations of a puzzle form a group, then in a sense all positions are the same - you can resticker any scrambled position to look like the solved position and no one would be able to tell the difference. For shape-changing puzzles, "restickering" would involve reshaping the external parts of the pieces, but the internal parts that form the mechanism remain untouched.

For bandaged puzzles however, the restickering only allows you to permute pieces of the same type. Even if you take all the stickers off, there is still a difference between the positions, due to the locations of the bandages blocking some moves and allowing others. Therefore there is no group underlying all the states of a bandaged puzzle, but if you restrict yourself to only those positions that have the bandages in the same place as the start position, and only use move sequences that return to that configuration, then you do have a group. Sliding piece puzzles similarly have an extra state, the location of the space, that needs to be factored out before it becomes a group.

The Nautilus is interesting in that respect, because the underlying group is trivial. It is not possible to permute pieces of the same size.

The Irreversible Cube also has some hidden state - the axis which was moved last and is therefore prevented from being moved next. This means that this puzzle really has three times as many positions as a normal 2x2x2 cube. It seems that it is not a group due to a different move being blocked each time, though you will get the ordinary 2x2x2 group if you restrict yourself to just those positions with the same 'bandaging' as the start position.

Luckily however the three solved positions are simply restickerings of each other. This suggests that we can still treat this larger set of positions as a single group, by always keeping this hidden state the same, and that does indeed work. Start with the UD axis being blocked. A move now consists of rotating the cube about the URF corner in either direction and doing a U move. This keeps the blocked axis the same so that you always have the same two moves available, and the group is generated by those two moves. You still cannot undo a move directly, but as there are only a finite number of positions, repeating a move will eventually return you to the same position and just before that last move you will have the position that is equivalent to having undone that move once.

Assuming my computer program is correct, if you start with one of the three solved positions, it takes up to 37 moves to reach any other position. If you start with any one of the three solved positions, then it takes at most 36.

The simplest (but not necessarily shortest) way to do a counterclockwise move is this:
[spoiler](FU)^15=I so U(FU)^14 = F'[spoiler]

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Fri Feb 22, 2013 3:11 am 
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bmenrigh wrote:
So does this puzzle form a group?


Yes. In fact it straightforwardly forms a permutation group. The two operations are that you can turn the top clockwise 90 degrees followed by rotating the whole puzzle counterclockwise 90 degrees while looking at the front face, and you can rotate the front face clockwise 90 degrees.

Clearly this puzzle qualifies as doctrinaire :-)

This raises the interesting question of whether it's possible to build this mechanism using the above property using gearing and ratcheting but no locking.


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Fri Feb 22, 2013 3:28 am 
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KelvinS wrote:
This should definitely be mass produced!
puzzle_weaver wrote:
This is at the top of my list of puzzles I want to see mass-produced.
Iranon wrote:
I sincerely hope it gets mass-produced at some point.
Thank you for the compliments for a design that hasn't even been prototyped yet. Mass-production is unlikely, given the current direction of the market. People buy Gear Cube because they understand what they buy. People recommend it to friends because they find that they can solve this difficult-looking (but easier than Rubik's) puzzle. Irreversible cube is the very opposite of an easily marketable puzzle.

Oskar

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Fri Feb 22, 2013 9:22 am 
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Oskar wrote:
People recommend [Gear Cube] to friends because they find that they can solve this difficult-looking (but easier than Rubik's) puzzle. Irreversible cube is the very opposite of an easily marketable puzzle.

I completely disagree with this. People buy puzzles that look intriguing but not too difficult. In fact they are completely put off buying puzzles that look too complex and difficult because they expect to fail to solve it and so won't bother buying and trying. But on the other hand, once they have bought a puzzle they are disappointed if they solve it too easily. The best way to exceed expectations is to provide a greater challenge than people expected, then they will recommend it to friends by saying it is a lot tougher than it looks. Those friends become even more intrigued and buy the puzzle to prove that it is not so difficult, and then their own expectations are exceeded, and so on.

This is exactly what happened with the Rubik's Cube, yet the marketers got it completely wrong by turning down the idea because it was too difficult, on the basis that "people want an easy challenge". In reality, people *think* they want an easy challenge (low risk of failing) *before* they buy a puzzle, but once they have bought it, they want the toughest challenge possible.

People don't buy puzzles that *look* too difficult for fear of failing, but they don't recommend them to friends if they solve them too easily, because they are ultimately disappointed.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Wed Apr 10, 2013 2:49 pm 
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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Wed Apr 10, 2013 3:22 pm 
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This is my favorite Oskar puzzle by far. It MUST be mass-produced!

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Wed Apr 17, 2013 7:08 pm 
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Wow... That is an incredibly EVIL puzzle! :twisted:

I'm also wondering if there's any way to cheat this puzzle. For example, on the Okamoto latch cube, it is possible to rotate a blocked face by inserting a credit card between the layers. This also has the unfortunate consequence of potentially creating a bricked or unsolvable puzzle.

However, with the Irreversible cube, I don't see any reason why forcing a blocked move once or twice would result in an unsolvable cube, since each position is functionally identical to every other position, quite unlike the myriad of other puzzles available with infinite varieties of mixed up parts, and the only thing potentially blocking a clockwise rotation on this cube is whether the preceding move was on the same face or not.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Mon May 06, 2013 12:00 pm 
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jaap wrote:
Assuming my computer program is correct, if you start with one of the three solved positions, it takes up to 37 moves to reach any other position. If you start with any one of the three solved positions, then it takes at most 36.

I think I don't understand what you mean with this.

There are three solved positions (the x-axis is locked, the y-axis is locked or the z-axis is locked).
Do you mean that, for example, when the x-axis is locked, you need 37 more turns to reach any other position. But when the y-axis or z-axis are locked you only need 36 to reach any other position?
How is that possible? Isn't a solved state with x-axis locked just a rotation of y-axis or z-axis locked?


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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Mon May 06, 2013 12:29 pm 
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pieppiep wrote:
jaap wrote:
Assuming my computer program is correct, if you start with one of the three solved positions, it takes up to 37 moves to reach any other position. If you start with any one of the three solved positions, then it takes at most 36.

I think I don't understand what you mean with this.

There are three solved positions (the x-axis is locked, the y-axis is locked or the z-axis is locked).
Do you mean that, for example, when the x-axis is locked, you need 37 more turns to reach any other position. But when the y-axis or z-axis are locked you only need 36 to reach any other position?
How is that possible? Isn't a solved state with x-axis locked just a rotation of y-axis or z-axis locked?


If, for example, you start with a solved cube that has the white face's axis locked, then there are some positions that need 37 moves to reach. Those exact positions will not need 37 moves if you had started from one or both of the other solved states. Of course for those other solved states there are positions 37 moves away that will be less if you started with the white face locked.

I'm not entirely sure if it is correct to turn this around. It may or may not be true that:
Every position can be solved in 36 moves if you don't care which of the three solved states you end up with, though you may need 37 moves if you want to end with a particular axis locked.

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 Post subject: Re: Oskar does it again! The Irreversible Cube
PostPosted: Sun Aug 04, 2013 3:32 am 
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I played with an Irreversible Cube at IPP, and solved it. A lot of fun -- so I am going home with it. :)

I'll write up something about my solution when I get a chance. IPP is very intensive, plus I am spending a lot of time writing programs for unbandaging Battle Gears / Gizmo Gears type puzzles. I hadn't seen this thread before, so started playing with it knowing nothing about it.


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