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 Post subject: Doctor SkewbPosted: Tue Mar 06, 2012 10:11 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Hello,

Last night I was breaking in another Doctor Skewb. I was trying NOT to scramble it too bad but the phone rang and I sat it down and when I got back to it I couldn't tell which slice I'd turned last so it ended up getting pretty well scrambled. I can solve the Doctor Cube even with the stickers on it so I figured I should at least be able to restore this to the cube shape. However after over an hour last night and a little more tinkering this morning this seems to be as close as I can get.

Attachment:

DrCube_Help.jpg [ 87.09 KiB | Viewed 2347 times ]

Just one edge pair is rotated by 90 degrees. Since this is a slice-turn-only puzzle the opposite edge looks just like this one. Could anyone give me a hand as to how best to get this back to a cube?

Thanks,
Carl

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 12:21 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Nice, this is why I love solving . Everything I know about this puzzle says that what you have should be impossible. I think the answer here is that the permutation of the pieces matters and since there aren't stickers we can't see how the permutation is wrong.

I have an idea but I can't follow it through to completion in my head so hang with me for a minute. First see these labels:
Attachment:

dr_cube_labels.png [ 378.61 KiB | Viewed 2311 times ]

First, ignore piece label D for a moment. I've only labeled one end of each of the piece groups because it should be enough to describe what I want you to try.

You can 3-cycle the A, B, and C pieces with a [1,1] commutator:

X = Move piece B to piece A's spot (ignore that piece A moves).
Y = Move piece C to the Piece A spot (which is occupied by piece B right now)
X' = Move the piece A spot piece (piece C) back to the piece B spot.
Y' = Move the piece A spot (piece A) back to the piece C spot.

This will cycle A -> C -> B -> A

My hope is that this cycle will do an odd number of 90 degree twists. Unfortunately I can't follow the orientations through in my head and there is a good chance that it does not do a odd number of 90 degree twists.

If after you apply that 3-cycle you're no closer to solved then undo it.

The other 3-cycle I want you to try uses piece D rather than piece C. That is, in the [1,1] commutator the B -> A slice is a 60 degree turn but the D -> A is a 120 degree turn. I'm pretty sure that will do an odd number of 90 degree twists.

Your goal here should be to either solve all of the orientations or end up with two edge groups twisted 90 degrees.

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 1:23 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
Nice, this is why I love solving . Everything I know about this puzzle says that what you have should be impossible. I think the answer here is that the permutation of the pieces matters and since there aren't stickers we can't see how the permutation is wrong.
Thanks! I'll give this a try when I get home tonight and let you know how it goes. Although I'm not sure even with stickers you couldn't get in the state with just one edge pair rotated 90 degrees. I was able to do it on the Doctor Cube.

Granted I couldn't really solve it directly from this state either but I just scrambled it up again and tryed again and I was able to solve it the second time. I avoided what ever parity issue that got me in that state.

Here (back talking about the Doctor Skewb) I could see something like the x-centers on one face all being rotated 90 degrees as a group (they'd still look solved) and just having a single edge pair rotated by 90 degrees. I'm just not sure. But I can assure you the above state IS possible. I put this together solved and it hasn't been taken apart. In fact taking it apart is risky as I've broken pieces doing that in the past... so that's why I'm here. I need to actually solve this before I can sticker it.

Carl

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 1:39 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
[...]Although I'm not sure even with stickers you couldn't get in the state with just one edge pair rotated 90 degrees. I was able to do it on the Doctor Cube.
I'm not 100% positive that you can't do it on a stickered Doctor Skewb but I have no idea how it would be possible. Obviously it is possible without stickers . I'd really be shocked if it was possible with stickers though.

The situation isn't a problem on the Doctor Cube though because the rotation of the centers isn't visible. In your case you just have an excess 90 degree twist in one of the centers. Speaking of the Doctor cube and interesting issues, you should be able to swap two edge-bars and have everything else solved. Have you run into that case yet?

On the Doctor Skewb though, all of the X centers that can be exchanged with edges have visible orientation. There aren't any tricky hidden orientations that allow the other 90 degree twist to go unseen.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 2:36 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
The situation isn't a problem on the Doctor Cube though because the rotation of the centers isn't visible. In your case you just have an excess 90 degree twist in one of the centers.
Agreed.
bmenrigh wrote:
Speaking of the Doctor cube and interesting issues, you should be able to swap two edge-bars and have everything else solved. Have you run into that case yet?
Not yet. I don't have a stickered one at the moment. Building one of those now as I sold the first one.
bmenrigh wrote:
On the Doctor Skewb though, all of the X centers that can be exchanged with edges have visible orientation. There aren't any tricky hidden orientations that allow the other 90 degree twist to go unseen.
Yes... and no. Yes, the X-Centers have visible orientation. But I guess you could say since there are 4 on each face with identical stickers they don't have a specific "position" in the solved state. One way I could envision to hide a 90 degree twist would be to rotate a group of 4 X-Center pairs about a face center by 90 degrees. See image below. Could a rotation of 1 to 2, 2 to 3, 3 to 4, and 4 to 1 such that the rest of the puzzle still appears solved be tied to a single edge pair rotation of 90 degrees? If not, what if 1 and 3 are swapped and each rotated 180 degrees?

Attachment:

DrCube_Help2.jpg [ 88.41 KiB | Viewed 2260 times ]

Carl

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 6:04 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
Yes... and no. Yes, the X-Centers have visible orientation. But I guess you could say since there are 4 on each face with identical stickers they don't have a specific "position" in the solved state. One way I could envision to hide a 90 degree twist would be to rotate a group of 4 X-Center pairs about a face center by 90 degrees. See image below. Could a rotation of 1 to 2, 2 to 3, 3 to 4, and 4 to 1 such that the rest of the puzzle still appears solved be tied to a single edge pair rotation of 90 degrees? If not, what if 1 and 3 are swapped and each rotated 180 degrees?
I had not thought about this possibility. I usually discount 4-cycles as impossible. On the Doctor Skewb 4-cyles are possible though (because a slice is a 6-cycle which is also odd. Unfortunately this 4-cycle changes the permutation parity and does not have any affect on orientation.

Also, I was totally wrong with my 3-cycle suggestion. With a lot more work I was able to track the twist and it doesn't pan out.

I'm pretty sure I (with a lot of help from DKwan) have a solution though! Hang with me on this, I've spelled out every detail to try to eliminate ambiguity. In the process, I've made this post long and complicated

First some notation:
Attachment:

dr_cube_transparent_notation.png [ 261.64 KiB | Viewed 2200 times ]

To describe sequences I will use Name, Direction, Amount:
R+60 is a 60 degree clockwise turn of the green slice. R-60 undoes it.

Observe that the the edges have a strange property that they LOOK like they have been twisted by 90 degrees when you flip their axis upside down by turning the slice 180 degrees. In this image the Atop and Abottom pieces will exchange places and the Btop and Bbottom pieces will also exchange places. Nothing else on the puzzle will move:
Attachment:

dr_cube_transparent_flipper.png [ 183.74 KiB | Viewed 2200 times ]

This sequence is a [1,1] orientation commutator:
F+180, R+60, F-180, R-60

It works by flipping A, then replacing it with B, then undoing the flipping (this time B), and then undoing the replacement.

When done it will LOOK like the A and B pieces have a 90 degree twist in them.

So how to we fix your single edge twisted? A trick . When an edge group is flipped it looks like it has been rotated by 90 degrees. When an X center is flipped upside down it looks like it has been twisted by 180 degrees

So take the above flipper commutator and change R+60 and R-60 to 30 degree turns:
F+180, R+30, F-180, R-30

Now (visually) the A piece is twisted by 90 degrees and the X center by 180. Using this trick you can transfer your 90 degree twist in the edge to a 180 degree twist in an X center.

Now you have to undo the 180 degree twist in the center. The way to do that is to do a 180 degree twist in an edge and X center since it isn't visible in the edge it will solve the puzzle.

Here is how to do that:
Attachment:

dr_cube_transparent_magic.png [ 165.03 KiB | Viewed 2200 times ]

There are two key observations to make before I describe the sequence. First, if you move the 1 piece around the corner from 1 -> 2 -> 3 -> 1 it will twist the 1 piece by 90 degrees. Second, the piece in the S spot is "safe" -- it doesn't move when you send the 1 piece around the corner.

So, the strategy is to construct an orientation changing commutator that manages to twist the S piece and the A piece while using the pieces in spots 1, 2, and 3 for the orientation changing portion of it.

To do this we're going to nest commutators. First, we need to get the A piece in the S spot with this 3-cycle:
L-60, F+30, L+60, F-30
This is a pure 3-cycle that does 1 -> A -> S -> 1

Now we need to twist the S piece in the 1 spot in place without breaking the A spot (the S spot will be preserved automatically).

To do that in words, we need to put 1 -> 2 slice (moves A too) and then 2 -> 3 slice and then undo the 1 ->2 which restores A, then 2 -> 3. Now the 1 spot is twisted and A and S are still intact.
In notation:
L+60, R+60, L-60, B-60

Now we undo the 3-cycle commutator:
F+30, L-60, F-30, L+60

Now we undo the orientation rotating:
B+60, L+60, R-60, L-60

Written out and grouped with [] the sequence is:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60], [F+30, L-60, F-30, L+60], [B+60, L+60, R-60, L-60]

Now, here is the trick. That will twist the 1 and S piece by 90 degrees. We need to twist them by 180. To do that we have to send the 1 spot piece around the corner TWICE:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60], [B+60, L+60, R-60, L-60]x2

The whole reason this twisted edge can happen is that without stickers a flipped piece looks the same as a twisted piece. With stickers you'd never end up in this situation.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 10:55 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
Now, here is the trick. That will twist the 1 and S piece by 90 degrees. We need to twist them by 180. To do that we have to send the 1 spot piece around the corner TWICE:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60], [B+60, L+60, R-60, L-60]x2
Now if I understand the notation correctly after this:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

I have this:
Attachment:

DrCubeH4.png [ 405.46 KiB | Viewed 2163 times ]

However that last:
[B+60, L+60, R-60, L-60]x2

Has no effect. I've gone through it twice. Just to make sure I understand when you say B+60 that is a 60 clockwise rotation about the B corner... correct? Because if you hold the puzzle with the F corner near you this appears to be counter-clockwise.

I feel I'm very close as I did manage to rotate that single face piece by 180 degrees (seen just in front of the B corner) and I suspect I should be able to take it from here but I'm really curious if I'm misunderstanding a bit of the notation of if there is a slight error here somewhere. I'll leave the puzzle in this state for a while so I can back track if I need to.
bmenrigh wrote:
The whole reason this twisted edge can happen is that without stickers a flipped piece looks the same as a twisted piece. With stickers you'd never end up in this situation.
Very very interesting... so I wonder if this means the Doctor Skewb is actually a harder puzzle to solve without it's stickers. I would have never guessed that.

THANKS!!!
Carl

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 Post subject: Re: Doctor SkewbPosted: Tue Mar 06, 2012 11:14 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
[...]I feel I'm very close as I did manage to rotate that single face piece by 180 degreesl
Indeed you are super close.

I think there was a tiny bit of miss communication. I intended the fix to be two steps and I didn't describe exactly how to chain them together.

I see that you applied that long
[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60], [B+60, L+60, R-60, L-60]x2
correctly.

It seems you didn't first transfer the 90 degree twist in the edge into an X center instead. I assumed you'd do that and then setup the pieces in whatever way was needed in order to make the second fix work. The good news is that the two sequences are commutative so you can do the first one now.

You just need to flip the twisted X center and the twisted edge upside down.

First you'll want to get them right next to each other so there is a setup move to do that:

R+120 which means turn slice around the R vertex 120 degrees clockwise. It should be right next to your twisted X center and between the B and L vertex.

Then you'll do the orientation flipping commutator:

R+180 (flips the twisted edge upside down)
F-30 (puts the flipped X center in that spot)
R-180 (flips the X center upside down
F+30 (puts the X center back in its original spot)

And then you can undo the setup moves slice with R-120.

So to solve the puzzle from where you are:
R+120, [R+180, F-30, R-180, F+30], R-120

I should order another print of the Doctor Skewb just for the stickerless challenge

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Doctor SkewbPosted: Wed Mar 07, 2012 1:18 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
I intended the fix to be two steps and I didn't describe exactly how to chain them together.
Ahh... makes sense. And just when I think I'm starting to understand this I get even more confused. Allow me to explain.
bmenrigh wrote:
So to solve the puzzle from where you are:
R+120, [R+180, F-30, R-180, F+30], R-120
Ok... I've now done this several times just to verify I wasn't making any errors. If I don't do the final R-120 I get this:
Attachment:

DrCubeH6.png [ 449.84 KiB | Viewed 2145 times ]

That last R-120 doesn't help as the entire R slice is already solved. I just have the one still incorrectly placed X-Center. Good news. From what I did above I know this will solve that.

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

It may be more then I need but I'm not sure where your fix went wrong. Let's go through it step by step.
bmenrigh wrote:
R+120 which means turn slice around the R vertex 120 degrees clockwise. It should be right next to your twisted X center and between the B and L vertex.
Done. Looks good.
bmenrigh wrote:
R+180 (flips the twisted edge upside down)
Done. Still good.
bmenrigh wrote:
F-30 (puts the flipped X center in that spot)
Done. Still good.
bmenrigh wrote:
R-180 (flips the X center upside down
Done... but I don't think this looks as you expect by this point. The opposite X-Center is in the exact same orientation as the one that was in this position before the R-180 turn. Flipping it upside down doesn't seem to change its apparent orientation.
bmenrigh wrote:
F+30 (puts the X center back in its original spot)
Done... but its not solved. Looks just like the one that was there.
bmenrigh wrote:
And then you can undo the setup moves slice with R-120
And at this point I'm really confused as the R slice is solved already before this R-120 move. If I were to do it that whole layer would just look 120 degrees off. Arg... and its getting late. I was thinking 180 degrees. 120 degrees shouldn't change the appearance of an already solved slice. I get it. It's just not needed here because the puzzle isn't stickered.

So your fix goes south with the R-180 move. But again I'm pretty sure that:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

will fix this.
bmenrigh wrote:
I should order another print of the Doctor Skewb just for the stickerless challenge
If I say yes you should does that look too self serving?

At this point I think the fix to the original problem would have been these 2 set up moves:

R+120, F+30

Followed by:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

And then undone the 2 setup moves with:

F-30, R-120

And 20 move sequence just to do what appears to be rotate an edge by 90 degrees. Wow! I don't feel so bad that I couldn't solve it. This is god's number for the Rubik's cube.

Carl

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 Post subject: Re: Doctor SkewbPosted: Wed Mar 07, 2012 3:24 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
bmenrigh wrote:
I intended the fix to be two steps and I didn't describe exactly how to chain them together.
Ahh... makes sense. And just when I think I'm starting to understand this I get even more confused. Allow me to explain.
bmenrigh wrote:
So to solve the puzzle from where you are:
R+120, [R+180, F-30, R-180, F+30], R-120
Ok... I've now done this several times just to verify I wasn't making any errors. If I don't do the final R-120 I get this:
[... image ...]
Carl I was certain you'd done something wrong but it turns out the answer to why this didn't work is actually the answer to your original question

Attachment:

dr_cube_x_180.png [ 449.84 KiB | Viewed 2133 times ]
If you do F+180 you flip this X center upside down and in the process flip mirror it front-to-back which looks like a 180 degree twist. The center has mirror symmetry along another plane though and when we pushed it over into the R slice with F-30 and then tried to flip it upside down with R+180, we mirrored it side-to-side. It looks same mirrored side-to-side which means the assumption that flipping it upside down would fix it was wrong.

That actually implies that your original problem of a single twisted edge could have been fixed in 4 moves:
R+180, L+30, R-180, L-30

I feel so stupid for not spotting which plane that pieces were being mirrored across or even realizing there was more than one possibility.

wwwmwww wrote:
That last R-120 doesn't help as the entire R slice is already solved. I just have the one still incorrectly placed X-Center. Good news.
Yes I was aware of that but I didn't want to leave it off because I think with it, the routine is a conjugate + commutator and without it, the routine is mostly a magical 5 moves. Also, if there was some other piece broken in the R slice, you'd have to do the R-120 or you'd see that the piece moved.

This is also why I've been saying R+180 and R-180 even though they are the same thing. One is the inverse of the other. If I were being flimsy with the notation I'd be providing mystery sequences (something I really dislike).

wwwmwww wrote:
bmenrigh wrote:
R-180 (flips the X center upside down
Done... but I don't think this looks as you expect by this point. The opposite X-Center is in the exact same orientation as the one that was in this position before the R-180 turn. Flipping it upside down doesn't seem to change its apparent orientation.
This was the key. I should have spotted it. Public solving humiliation at its finest.

wwwmwww wrote:
[...]But again I'm pretty sure that:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]
will fix this.
Yep it does but for a different reason.

Also, the reason the [B+60, L+60, R-60, L-60]x2 is not needed at the end is that when applied twice, the net effect is a 120 degree twist in both the B and R slices but since you're at a nearly solved state you can't see those. The whole "fix" with the last part of the commutator gives you the ability to twist two pieces by 90 or 180 degrees, even on a Stickered Doctor Skewb .

wwwmwww wrote:
bmenrigh wrote:
I should order another print of the Doctor Skewb just for the stickerless challenge
If I say yes you should does that look too self serving?
Nope! I just ordered another print I'll dye it black and either put white stickers or maybe mirror stickers. It's a totally different puzzle and apparently a really fun one too!

wwwmwww wrote:
At this point I think the fix to the original problem would have been these 2 set up moves:

R+120, F+30

Followed by:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

And then undone the 2 setup moves with:

F-30, R-120

And 20 move sequence just to do what appears to be rotate an edge by 90 degrees. Wow! I don't feel so bad that I couldn't solve it. This is god's number for the Rubik's cube.

Carl
So I don't think that'll do it. That'll twist the broken edge by another 180 degrees making 270 which just looks like a 90 degree twist.

I will definitely be making a video of what is going on as soon as I can.

I'm glad you got the puzzle solve. I was a really fun mental challenge for me even though I went about the solution in such a round-about way that I wasted a lot of your time in the process...

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: Doctor SkewbPosted: Wed Mar 07, 2012 10:18 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
That actually implies that your original problem of a single twisted edge could have been fixed in 4 moves:
R+180, L+30, R-180, L-30
Oh... Nice. I'll need to try this when I get home.
bmenrigh wrote:
This is also why I've been saying R+180 and R-180 even though they are the same thing. One is the inverse of the other. If I were being flimsy with the notation I'd be providing mystery sequences (something I really dislike).
No problem. I appreciate you being so careful with the notation. This will come in handy after I have it stickered.
bmenrigh wrote:
This was the key. I should have spotted it. Public solving humiliation at its finest.
No humiliation intended. I actually like the way this turned out. I learned more. I think this is a clear case where seeing what doesn't work and why teaches you just as much, if not more, then seeing what does work.
bmenrigh wrote:
wwwmwww wrote:
[...]But again I'm pretty sure that:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]
will fix this.
Yep it does but for a different reason.
So is this the best way to twist a singe X-Center pair on an unstickered Doctor Skewb? Can this be shortened to 4 moves as appears to be the case for a single twisted edge? And it also make me wonder if its possible to twist a single X-Center pair by 90 degrees without appearing to change anything else on an unstickered Doctor Skewb.
bmenrigh wrote:
Nope! I just ordered another print I'll dye it black and either put white stickers or maybe mirror stickers. It's a totally different puzzle and apparently a really fun one too!
NICE!!! Hmmm... Does OlivĂ©r offer mirror stickers? I need to check.
bmenrigh wrote:
So I don't think that'll do it. That'll twist the broken edge by another 180 degrees making 270 which just looks like a 90 degree twist.
Are you sure? I guess I need to test it. I thought it was the X-Centers that when flipped appeared to be rotated by 180 degrees but the Edges only appeared to be rotated 90 degrees. Oh wait... does this sequence actually flip the piece? Maybe its actually rotating the piece, in which case I think you are right. So much is hidden about what is actually going on when the puzzle isn't stickered. Yes, this is quite fun.
bmenrigh wrote:
I will definitely be making a video of what is going on as soon as I can.
Looking forward to it.
bmenrigh wrote:
I'm glad you got the puzzle solve. I was a really fun mental challenge for me even though I went about the solution in such a round-about way that I wasted a lot of your time in the process...
Trust me you didn't waste my time. I enjoyed every minute. Well aside from the fact that I'm still breaking in the puzzle and its turning isn't the best yet. But I can't think of a better way to break it in. Oh... and aside from being difficult to turn occassionally keeping track of the F, L, R, and B corners while executing a 20+ move sequence when the puzzle isn't stickered is a challenge just by itself. Half way through this:

[L-60, F+30, L+60, F-30], [L+60, R+60, L-60, B-60]x2, [F+30, L-60, F-30, L+60]

This get to look very scrambled so if you lose sight of which corner is which I think you could be lost in no time.

I had a blast. THANKS AGAIN!!!
Carl

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 Post subject: Re: Doctor SkewbPosted: Wed Mar 07, 2012 9:15 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
bmenrigh wrote:
That actually implies that your original problem of a single twisted edge could have been fixed in 4 moves:
R+180, L+30, R-180, L-30
Oh... Nice. I'll need to try this when I get home.
Just tested this and YES it works.

Thanks again,
Carl

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 Post subject: Re: Doctor SkewbPosted: Sun Mar 11, 2012 2:46 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Thanks to this thread you can now order Mirror Stickers for your Doctor Skewb here.

Carl

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