# Maze Cube solutions



## Johannes91 (Feb 1, 2010)

Interesting facts:

- It's possible to swap two edges without changing anything else.
- It's *not* possible to swap two corners without changing anything else.
- Every position has a "complement" that you get by rotating every sticker with a straight line 90 degrees and every sticker with a straight angle 180 degrees.

Below is the distribution of solved states with different numbers of loops. They are divided to 6 subgroups depending on parities. *[Edit: This is a bit wrong, see Jaap's post below.]*


```
Loops: |   1 |   2 |   3 |   4 |   5 |   6 | Total 
-------+-----+-----+-----+-----+-----+-----+-------
    G0 |  16 |  28 | 152 |  22 |  78 |   0 |   296 
    G1 |  20 |  38 | 136 |  20 | 106 |   0 |   320 
    G2 |  25 |  39 | 150 |  33 | 105 |   4 |   356 
    G3 |  35 |  55 | 140 |  43 | 105 |   4 |   382 
    G4 |  25 |  39 | 150 |  33 | 105 |   4 |   356 
    G5 |  35 |  55 | 140 |  43 | 105 |   4 |   382 
-------+-----+-----+-----+-----+-----+-----+-------
Total: | 156 | 254 | 868 | 194 | 604 |  16 |  2092
```

All solved states in awesome ASCII art: maze.txt (5 MB) or maze.txt.gz (0.2 MB). The numbers before each cube are the number of loops and the subgroup.

I am fairly confident my program works correctly, but it's possible that some solutions are missing, some subgroups are wrong, or I screwed up in some other way. Has this been done before? The little googling I did turned up next to nothing.


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## jiggy (Feb 1, 2010)

> - It's possible to swap two edges without changing anything else.
> - It's *not* possible to swap two corners without changing anything else.



Hi, Johannes. I may be wrong (in fact I'm frequently wrong!) but if the maze cube is like the normal 3x3 I think it _is_ possible to swap two corners without swapping anything else:

[(L' U' L) D' (L' U L) D] [(F' R' F) L (F' R F) L']

Of course, both corners actually stay in the same position and just change orientation, but I think the same is true for swapping two edges, right? I don't think it's possible to cycle only two edges, even if you do flip them, as those are two separate cube laws. Have I made some n00bish mistake here?

I don't know if this changes anything for you, but I thought I'd mention it! =)


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## rubixfreak (Feb 1, 2010)

SO what if i manage to swap UR with UL without changing the rest - what you claimed is possible
and than apply an T-permutation?
the edges will get swapped back so will be solved and the UFR and UBR corner will get swapped. so we end up with a solves cube with two corners swapped.
->your assumptions must be wrong


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## Johannes91 (Feb 1, 2010)

jiggy: Your alg _twistes_ two corners. By swapping I meant a 2-cycle. Not n00bish at all, just a terminology mismatch.

There is a way to swap 2 edges on Maze Cube, even though it's impossible on a normal 3x3x3. I can post an alg and explanation later, but it's a really nice puzzle and I don't want to spoil it yet.

rubixfreak: Good try, but do you have a T-permutation alg that doesn't rotate any centers?


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## Stefan (Feb 1, 2010)

rubixfreak said:


> ->your assumptions must be wrong


You're talking to Johannes.
->your understanding of him must be wrong


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## jiggy (Feb 1, 2010)

Johannes91 said:


> jiggy: Your alg _twistes_ two corners. By swapping I meant a 2-cycle. Not n00bish at all, just a terminology mismatch.
> 
> There is a way to swap 2 edges on Maze Cube, even though it's impossible on a normal 3x3x3. I can post an alg and explanation later, but it's a really nice puzzle and I don't want to spoil it yet.



When you say the Maze Cube, is this the kind of thing you are referring to?

http://www.puzl.co.uk/images/MZ.jpg

From the picture, it looks just the same as a 3x3. What is it about the maze cube which allows you to cycle two edges?


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## Stefan (Feb 1, 2010)

jiggy said:


> When you say the Maze Cube, is this the kind of thing you are referring to?
> 
> http://www.puzl.co.uk/images/MZ.jpg


Yeah. Here it is in 3D:
http://www.randelshofer.ch/cubetwister/doc/cubes/maze_cube.html



jiggy said:


> From the picture, it looks just the same as a 3x3. What is it about the maze cube which allows you to cycle two edges?


C'mon, that's the interesting part of the riddle...


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## PHPJaguar (Feb 1, 2010)

Johannes91 said:


> rubixfreak: Good try, but do you have a T-permutation alg that doesn't rotate any centers?


Why do the centers matter?


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## Muesli (Feb 1, 2010)

PHPJaguar said:


> Johannes91 said:
> 
> 
> > rubixfreak: Good try, but do you have a T-permutation alg that doesn't rotate any centers?
> ...


Because it's a maze cube, and centres have a permutation.


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## Johannes91 (Feb 1, 2010)

PHPJaguar: Rotating a center is a visible effect that changes the position, just like flipping an edge or twisting a corner. (Two centers have a straight line on them and four have a straight angle.)

How to swap two edges, for those who don't want to figure it out:



Spoiler



The key is that there are many identical centers.

Do this on a 3x3x3: E M u M' U' M' U' M' U2 M U M' U M U M' U M U'.

It swaps the edges UL-UR and centers U-L, D-R, and F-B. What if all the swapped center pairs were identical? The only visible effect would be the UL-UR edge swap!


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## Stefan (Feb 1, 2010)

Spoiler



Bah... I posted a shorter one years ago 
http://games.groups.yahoo.com/group/blindfoldsolving-rubiks-cube/message/464
Yours might very well be faster, though.


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## jiggy (Feb 1, 2010)

StefanPochmann said:


> jiggy said:
> 
> 
> > From the picture, it looks just the same as a 3x3. What is it about the maze cube which allows you to cycle two edges?
> ...



Haha, I think I may have figured it out, is the answer basically this?



Spoiler



M S M' U R U R' U' M' U R U' r' U' x2 U R U R' U' M' U R U' r' U' D M U2 M' U R2 U R U R' U' R' U' R' U R' U D'

(I'm _sure_ that that isn't optimal but hopefully you get what I mean!)



EDIT: Looking at your answers, I can see I'm thinking along the right lines here.


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## Stefan (Feb 1, 2010)

Now that you've looked already: When you QUOTE someone, you get to see their message source code, so you can see how we did the spoilers.

And yes, you're going the right direction, although yours doesn't quite work for the actual maze cube pattern.


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## Johannes91 (Feb 1, 2010)

Stefan:


Spoiler



Your alg is definitely much cleaner. The way I found mine was E M u --> use a <M,U>-solver.  Just wanted something that works to show the position.



jiggy:


Spoiler



Yeah, that's the general idea, but your exact alg can't be used because there's no slice with 4 identical centers. (The two centers with a straight line are adjacent to each other, not opposite.) This might not be obvious if you don't have a real maze cube and haven't looked at the stickering too closely.


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## jiggy (Feb 1, 2010)

StefanPochmann said:


> Now that you've looked already: When you QUOTE someone, you get to see their message source code, so you can see how we did the spoilers.
> 
> And yes, you're going the right direction, although yours doesn't quite work for the actual maze cube pattern.


Haha, that's a really obvious tag...you would have thought I would have been able to figure that out for myself! Sorry, I guess I'm still getting used to the forum. 



Johannes91 said:


> jiggy:
> 
> 
> Spoiler
> ...


Ah, I see. Thanks a lot for the explanation, I feel like I've learnt a lot today!


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## Stefan (Feb 1, 2010)

Or edit the text of your link: my solution
Actually it's yours, I just want to show how it could've looked.


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## Lucas Garron (Feb 1, 2010)

I vote for this alg:



Spoiler



M' u M' U M' U' M' U' M' U2' M' U' M' u'


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## jaap (Feb 2, 2010)

Johannes91 said:


> ```
> Loops: |   1 |   2 |   3 |   4 |   5 |   6 | Total
> -------+-----+-----+-----+-----+-----+-----+-------
> G0 |  16 |  28 | 152 |  22 |  78 |   0 |   296
> ...



I did this about 8 years ago, and also found 2092 solutions. 
http://www.twistypuzzles.com/forum/viewtopic.php?f=1&t=4266&p=27569#p27569.


```
Orbit  Total 1loop    2    3    4    5    6
2a0      346    29   45  137   35   96    4
2a1      392    31   49  153   41  114    4
1a0      346    29   45  137   35   96    4
1a1      392    31   49  153   41  114    4
0a0      320    24   34  136   24  102
0a1      296    12   32  152   18   82
Total   2092   156  254  868  194  604   16

2o0       62     3    2   25    3   29
2o1       67     4    7   24    3   29
1o0       62     3    2   25    3   29
1o1       67     4    7   24    3   29
0o0       92     6    6   36    4   40
0o1      110     6   12   42    4   46
Total    460    26   36  176   20  202    0
Total   2552   182  290 1044  214  806   16
```

I get the same totals, but one of us has misidentified which orbit some of the patterns fall into. I have also listed the patterns where the straight line centres are on opposite faces. See the twistypuzzle forum post for the complete list of patterns.


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## Johannes91 (Feb 2, 2010)

jaap: Thanks! I thought I had seen that before, but couldn't remember where and my google-fu failed me. Should've guessed it was you.

I found a silly bug in my code which made it count the center rotations wrong. It agrees with your results exactly now.


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