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 Post subject: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 10:24 am 
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I'm new to the whole idea of restickerings/recolorings of the rubik's cube (so far all I've done is place various oriented/colored/not pieces in the empty spot in my void cube) but I have been wondering, is there a maximally difficult coloring of a given puzzle? On a 3×3×3 at least, the "photo cube" coloring is not the most difficult since leaving the centers identically blank causes a parity to arise. But a void cube is not the most difficult either since one does not have to orient centers on a void cube! It's possible to combine these two specific issues of course, by having identically colored centers but with a correct orientation (one way would be having a black line connecting the center to one adjacent piece; another way would be to put a number and arrow on each center, requiring the arrow point towards the next higher number, though I suppose on such a puzzle one could memorize which number should match which color to avoid parity).

So, to summarize; sometimes information can be added to the coloring to make an extra thing to solve for; sometimes information can be taken away from the coloring to make cases occur which normally wouldn't; and sometimes it's possible to have the best of both worlds. So what I'm wondering is whether it's always possible; is there a Rubik's cube with all the difficulties of any one cube stickering, and does this generalize to other twisty puzzles?


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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 10:32 am 
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Correct me if I'm wrong, but the solution with the black lines connecting the centers to an adjacent edge would, in fact, be the solution with the highest number of possible positions. But that doesn't necessarily make it the most difficult, at least from my experience. It's slightly harder, as it requires an additional algorithm to rotate a center piece 180 degrees, but I actually find it quite easy to solve compared to variants like the Shepard's cube.


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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 10:37 am 
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I don't think there is such a thing a "maximally difficult" because each person finds different things difficult and some of them are conflicting.

An a Rubik's cube there's a list of things that can't happen:

* A single flipped edge
* A single twisted corner
* Corners solved but a single pair of edge swapped (and vice versa)
* A single center twisted 90 degrees (related to the swapping of edges and corners)

By playing with the stickers you can cause some of these problems.

To allow a single flipped edge, take the stickers of an edge.
To allow a single twisted corer take the stickers off a corner.
To decouple the twist of the centers from the parities of th edges and corners make there two identical edges and two identical corners. It would suffice to take the stickers off of a second edge and second corner so that there are two blanks of each.

The problem with saying this is "maximally hard" is that more information can be harder (think picture cube) or less information can be harder (think taking off stickers as suggested above). A pretty hard color scheme would be one that manages to show the orientation of the centers and stickers on most pieces without making it obvious which pieces are the modified ones. If the solver knows which pieces are the troublemakers they can usually save them until the end where the problems won't be obvious (two flipped blank edges doesn't look like an issue).

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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 11:05 am 
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bmenrigh wrote:
The problem with saying this is "maximally hard" is that more information can be harder (think picture cube) or less information can be harder (think taking off stickers as suggested above). A pretty hard color scheme would be one that manages to show the orientation of the centers and stickers on most pieces without making it obvious which pieces are the modified ones. If the solver knows which pieces are the troublemakers they can usually save them until the end where the problems won't be obvious (two flipped blank edges doesn't look like an issue).


What I'm aiming at is more the mathematical abstraction. One could 'color' a rubik's cube with numbers and ask the solver to put only numbers with a common divisor on a given face, but do it such that it would be equivalent to solving an ordinary Rubik's cube once the player got used to which of the chosen numbers shared factors. (Or this puzzle could be made to be quite ambiguous, possibly with multiple solutions...) So I think the (interesting) problem of how best to implement a coloring can be separated from the mathematical abstraction of which positions (of individual cubies) we are calling 'solved'.

However, maybe I am wrong and they can't really be considered separately. The way I'm thinking of it is 'just making things confusing doesn't count, the same information is there' but I suppose if it were really possible to immediately absorb all the information a coloring presents, we would all be able to solve cubes blindfolded. And your example of leaving the de-stickered pieces for last is a good one. Yet, I am uncertain.

I wonder if there is a coloring of the void cube which makes it obvious ahead of time whether you will run into a parity. My understanding is that you could predict parity ahead of time by counting the number of mis-oriented pieces, is that correct? I guess anything can be predicted ahead by just solving it in your head.

Quirky-Cubes wrote:
Correct me if I'm wrong, but the solution with the black lines connecting the centers to an adjacent edge would, in fact, be the solution with the highest number of possible positions. But that doesn't necessarily make it the most difficult, at least from my experience. It's slightly harder, as it requires an additional algorithm to rotate a center piece 180 degrees, but I actually find it quite easy to solve compared to variants like the Shepard's cube.


Adding a black line reduces the number of possible positions, so I'm not sure what you mean.

Unfortunately I've never played with a Shepherd's cube, though I'm familiar with them. Is the difficulty created by the presentation of the information or by actual ambiguity?


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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 11:57 am 
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If we are speaking about the maximum number of position, then the super cubes are the one you are searching for. (It can be something from Pochman style, to photo cube)
If you are looking for something what is really hard to solve, I recommend the Maze cube:
http://www.randelshofer.ch/rubik/virtua ... _cube.html

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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 12:19 pm 
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dranorter wrote:
Unfortunately I've never played with a Shepherd's cube, though I'm familiar with them. Is the difficulty created by the presentation of the information or by actual ambiguity?
The Shepherd's cube (arrow cube) is hard because it has duplicate corners and duplicate edges. Also, there is a corner where the direction of the arrows makes it so that you can't see the orientation of the piece.

The number of permutations of a puzzle is only loosely correlated with the difficulty of the puzzle. The same goes for sticker mods. I don't think there is an objective mathematical way to capture this.

For example, restoring the original color scheme of a 10-color Dogic is a lot harder than a 12-color Dogic, despite the Wikipedia article saying "The 12-color Dogic is the most challenging version". Using the number of permutations of the sticker variation as a measure of difficulty may be a good mathematical approach but it isn't a good measure of human solving difficulty.

Here is another example. If you sticker a Helicopter cube like this:
Image
It will have roughly 3^7 / 24 fewer permutations but many people will find it much harder because the coloring of the central triangles interacts with the path of the orbits to create a hidden orientation problem.

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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 1:55 pm 
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Olivér Nagy wrote:
If we are speaking about the maximum number of position, then the super cubes are the one you are searching for. (It can be something from Pochman style, to photo cube)
If you are looking for something what is really hard to solve, I recommend the Maze cube:
http://www.randelshofer.ch/rubik/virtua ... _cube.html


Agree. Maze cube, sodoku cube, snake cube etc are far harder than the regular cubes, given that you don't know the final state (well, you shouldn't know it).


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 Post subject: Re: Is there such a thing as a maximally difficult coloring?
PostPosted: Mon Feb 13, 2012 1:59 pm 
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bmenrigh wrote:
The number of permutations of a puzzle is only loosely correlated with the difficulty of the puzzle. The same goes for sticker mods. I don't think there is an objective mathematical way to capture this.


It is my conviction that there are almost always mathematical formulations for intuitively defined concepts. It just may take decades or centuries for people to find them.

I certainly am not proposing that number of permutations is the correct measure! If I were to make a proposal it would be more like, number of looks required during a solve, or the amount of memory required to solve maybe (obviously if the memory is more than the cube size you can always solve). One would hope that some algorithmic consideration like this would turn out to line up with a more abstract mathematical property of a given coloring. Then one could begin to really start proving theorems. :) Human ease-of-solving would always have some idiosyncrasies as compared with theoretical ease-of-solving, but mostly it would boil down to maybe two or three different theoretical measures (like memory or # of looks).

To me it seems there is a constant dance of exploration, every new puzzle or coloring suggests more possibilities; some simply explore the mathematical space available whereas others redefine it (say, whoever was first to use nonplanar cuts or nonconvex solids or jumbling puzzles). This makes it seem like there is no single mathematical framework encompassing the area but that might be just because we haven't explored enough yet to see the big picture; for example before the Church-Turing thesis people were making lots of different definitions of "algorithm" and didn't know that there could be one sufficiently general definition to encompass them all.

schuma wrote:
Olivér Nagy wrote:
If we are speaking about the maximum number of position, then the super cubes are the one you are searching for. (It can be something from Pochman style, to photo cube)
If you are looking for something what is really hard to solve, I recommend the Maze cube:
http://www.randelshofer.ch/rubik/virtua ... _cube.html


Agree. Maze cube, sodoku cube, snake cube etc are far harder than the regular cubes, given that you don't know the final state (well, you shouldn't know it).


Heh, I went and tried to solve the maze cube, my solution was to draw what the final configuration would be before scrambling. :) Maybe I should be more open-minded though, it makes perfect mathematical sense to suppose a solver doesn't know the goal configuration, and in fact works ok as a model of an ordinary Rubik's cube.


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