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 Post subject: Very Strange SubgroupsPosted: Mon Sep 23, 2013 11:03 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Starting a new thread to stop derailing the one about the Incomprehensible Cube, where this very interesting subject accidentally came up. The story so far is that we have two very odd subgroups which haven't to date been studied, specifically (UD Fs Ls) and (UD L R). For both of these, we have the questions:

What is up with the bizarro number of positions in the subgroup?

How would you go about solving them?

How would you go about building them?

Are there other subgroups with similar interesting properties?

I'm particularly interested in this, because it's so rare to come across solving experiences which aren't rehashings of the same old ABA-B- tricks, and most of the remaining ones are like the gear shift, which has simple linear components. It's rare to encounter something truly new, and usually when you do they're very simple, like the alternating cube, which is our most recent truly novel solve.

I'll start with (UD Fs Ls). This one is very confusing until you make the deep observation that it's equivalent to (UsU2 Fs Ls). This same insight shows up in the Gear Cube as well, which I solve by getting it to the subgroup which can be solved using (RsR2 FsF2) and then it seems to magically solve itself. (I'd use LsL2, but I'm right handed. Apparently Andreas is left-handed, because he likes saying Ls.)

Once you make the (UsU2 Fs Ls) observation, it because straightforward to understand what's going on. The horizontal edges have four possible positions. Each of the other two bands of edges is a 2x2x1, which has 6 positions, times 2 because it can be flipped over and times 4 because it can be oriented, and the face centers are a cube orientation which can be in 24 possible positions. Since all four of those must have the same parity you divide by 8 in the end, which gives a total number of positions of 4 * (6 * 2 * 4)^2 * 24 / 8 = 27648.

Solving it is quite simple once you understand what's going on. First you orient the horizontal edges properly, then you solve the Fs using the (UsU2 Fs) subgroup, which is easy once you realize the band is really a 2x2x1, then you solve the Ls band similarly, and finally you position the centers using FsLsFs-Ls- and the like.

It's straightforward to build using a 1x1x3 geared core, with the horizontal edges and all corners glued directly to the core, and the other faces and edges being tiles which float around like in the mixup cube. Which raises the question of what this is like in a mixup version. I suspect not so interesting, but that's off the subject.

(UD L R) appears to be much stranger. Andreas will have to chime in with the size of its subgroup, but given other examples it's probably the size of (U D L R) divided by 8505. If it's something less interesting then (UD L) is probably the thing to study. I have no idea to make of the strange 8505 = 3^5 * 5 * 7 number.

It can be built quite simply by gearing together the top and bottom axes through the central core of a regular Rubik's Cube mechanism, and gluing the front and back centers in place.

For other puzzles worth exploring for similar properties, it's probably best to look at axis flipping variants. The variants of (UD Fs Ls) such as (UD FB- LR-) are probably essentially similar to the first one, but the variants of (UD L R) such as (UsU2 L R) and (UD Ld Rd) might be very interesting, albeit mindbending and technically very challenging to build. (d is for deep, so Ld is just shorthand for LLs).

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 Post subject: Re: Very Strange SubgroupsPosted: Tue Sep 24, 2013 9:52 am

Joined: Thu Sep 17, 2009 6:07 am
Location: Germany, Bavaria
Andreas, let me continue here and ask again: What do I need to run GAP on Windows?

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 Post subject: Re: Very Strange SubgroupsPosted: Tue Sep 24, 2013 10:02 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Just to keep track for future reference:
viewtopic.php?f=15&t=26068
(the thread where this discussion started)
Bram wrote:
(UD L R) appears to be much stranger. Andreas will have to chime in with the size of its subgroup, but given other examples it's probably the size of (U D L R) divided by 8505. If it's something less interesting then (UD L) is probably the thing to study. I have no idea to make of the strange 8505 = 3^5 * 5 * 7 number.
Code:
gap> Size(Group(U*D, L, R));
119439360
gap> Size(Group(U, D, L, R));
21119142223872000
gap> Size(Group(U, D, L, R))/Size(Group(U*D,L,R));
176818950
gap> 176818950/8505;
20790
gap> Size(Group(U*D, L));
9953280
BTW: I am right-handed but germans write from left to right.

You find the GAP-system here:
and a quick tutorial for Rubiks Cube here:
http://www.gap-system.org/Doc/Examples/rubik.html
For analyzing our subgroups the definition of Rubiks Cube given there is not suitable. You have to include the face pieces either with or without orientation. In both cases you can use the attached file further down. You should give it the extension ".g"

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 Post subject: Re: Very Strange SubgroupsPosted: Tue Sep 24, 2013 11:01 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I figured out where the 8505 came from. When the top and bottom were put in lock step, it made it so that each individual corner can still go to any position, but each corner has a twin which it moves in lock step with, which resulted in the number of possible corner positions going from 8 * 7 * 6 * 5 * 4 * 3 * 2 to 8 * 6 * 4 * 2, for a ratio of 7 * 5 * 3. Further, four of the corners were made to be in lock step with their pairs, for a further reduction of 3^4, so the total ratio is 7 * 5 * 3 * 3^4, or 8505.

Clearly it's piece pairing which causes most of the interesting stuff to happen, and (UD L R) is the thing to study because it does that for both the corners and edges. Which still leaves open the question of what the useful sequences are for solving it.

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 Post subject: Re: Very Strange SubgroupsPosted: Wed Sep 25, 2013 1:51 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Some more progress on analyzing (UD R L). The corners and edges come in pairs, so there are for the edges 6! positions for the pairs as a whole and it's possible to flip the positions of just the two pieces in one pair, so that's 6! * 2^12. Likewise the corners have 4! * 2^8, and there are 3^4 orientations of the corners, so that yields 6! * 4! * 2^20 * 3^4 = 1433272320 which is a factor of 12 high. One factor of 2 is inherited from the Rubik's Cube. Presumably a factor of 3 is a limitation on corner orientations. The other factor of 2 I have no idea. Can anybody analyze this further?

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 12:55 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Come to think of it, the piece pairing property also applies to (UD FB R L). "But Bram" I hear you saying "Isn't that just the Gear Cube Extreme?" It is not! The Gear Cube Extreme is an axis flipping puzzle! It corresponds to (UsU2 FsF2 R L) and has completely different properties. Silly me to forget that there was another perfectly reasonable set of twins for which one of them is already in mass production!

We seem to have a very surprising general principle, which is that axis-flipping antislice moves much more natural from a solving and mechanism designing standpoint than plain shallow cut antislice moves.

So, looks like we now have a clear winner as to what strange puzzle should be built/solved - the Gear Cube Extreme Evil Twin. Who's going to take a stab at it?

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 2:33 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
So, looks like we now have a clear winner as to what strange puzzle should be built/solved - the Gear Cube Extreme Evil Twin. Who's going to take a stab at it?
Not certain but I think I ALREADY have this one. Isn't this just item 6 on my Multi Gear Cube Kit.

There I call it the Anisotropic Cube which I think is just another name for the Gear Cube Extreme but now I'm not so sure that really is what Item 6 on this list is. And I now also think Item 2 on my list isn't the same puzzle as Even Less Gears.

I've sold my first Gear Cube Kit and I now have a second one I need to dye and assemble so maybe (UD FB R L) is the first state I should put that one together in.

Carl

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 3:37 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Carl, the number of possible positions you have listed for the 'Anisotropic Cube' (apparently actually its doppelganger) appears to be correct, although I'm not sure what the mechanism of your multi gear kit is. Are there any pictures/videos of it assembled in the anisotropic configuration?

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 4:08 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
Carl, the number of possible positions you have listed for the 'Anisotropic Cube' (apparently actually its doppelganger) appears to be correct, although I'm not sure what the mechanism of your multi gear kit is. Are there any pictures/videos of it assembled in the anisotropic configuration?
Bram,

http://twistypuzzles.com/forum/viewtopic.php?f=15&t=23935

Here are the videos I currently have:

I have not yet assembled it in that configuration but it would require only 8 of the 12 1:1 Gears which come with the kit. And the one you see in that thread and the videos was sold at last years IPP. I've have since made Gear Cube Kit version 2.0 and its still currently in the fresh from Shapeways state. I should have it ready to assemble in a week or two and this is the first state I'll put it in.

As for the above table... Andreas did the calculations for the number of states and at the time I was personally thinking this was the same puzzle as the 'Anisotropic Cube' aka 'Gear Cube Extreme' so I'm not certain if Andreas did the correct math and I just had the wrong picture in my head or if my incorrect picture influenced Andreas' math. What is the number of states of the 'Anisotropic Cube' aka 'Gear Cube Extreme'? If I've followed this correctly I think you are saying the number of states in the table above is the correct number for the doppelganger... correct? If so then I'd say Andreas's math was unswayed by my poor understanding at the time.

Carl

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Last edited by wwwmwww on Thu Sep 26, 2013 5:46 pm, edited 1 time in total.

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 5:20 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Yes, the number of states in the chart above is correct, and that configuration you have labelled Anisotropic is most definitely NOT isomorphic to the Anisotropic/Gear Extreme cube by Oskar. It's the doppelganger, which is a much more novel solve experience.

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 Post subject: Re: Very Strange SubgroupsPosted: Thu Sep 26, 2013 6:07 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
Yes, the number of states in the chart above is correct, and that configuration you have labelled Anisotropic is most definitely NOT isomorphic to the Anisotropic/Gear Extreme cube by Oskar. It's the doppelganger, which is a much more novel solve experience.
Items 17 and 19 on the list also have the same number of states as this doppelganger (at least as long as you aren't looking at face center orientation). I suspect they may be even more interesting from a solving perspective due to the additional restriction on the 3rd axis. Not certain about that but thought I'd throw that out there to see if you agreed.

Carl

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 Post subject: Re: Very Strange SubgroupsPosted: Fri Sep 27, 2013 4:53 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Just for the sake of completeness:
Code:
#The doppelganger
gap> Size(Group(U,D,L*R,F*B));
3822059520
#The Gear Cube Extreme
gap> Size(Group(U,D,L*L*Ls,F*F*Fs));
2497078886400

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 Post subject: Re: Very Strange SubgroupsPosted: Fri Sep 27, 2013 9:01 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
Just for the sake of completeness:
May I ask what the size is of Even Less Gears? Can it even be expressed in your notation? I'm not certain this is even a doctrinaire puzzle... if it is its not obvious.

Thanks,
Carl

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 Post subject: Re: Very Strange SubgroupsPosted: Fri Sep 27, 2013 10:53 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
wwwmwww wrote:
Items 17 and 19 on the list also have the same number of states as this doppelganger (at least as long as you aren't looking at face center orientation). I suspect they may be even more interesting from a solving perspective due to the additional restriction on the 3rd axis.

Both of those are very similar, but lose the nice property that each axis is individually symmetric. Item 19 is probably only a little bit more difficult, because it's blocking one face from moving. 17 is probably obnoxiously difficult, because it has some of that pasa doble property. I think the Anisotropic Doppelganger is a much more canonical puzzle.

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 Post subject: Re: Very Strange SubgroupsPosted: Fri Sep 27, 2013 10:57 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
wwwmwww wrote:
May I ask what the size is of Even Less Gears? Can it even be expressed in your notation? I'm not certain this is even a doctrinaire puzzle... if it is its not obvious.

Even less gears is not a doctrinaire puzzle.

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 Post subject: Re: Very Strange SubgroupsPosted: Fri Sep 27, 2013 11:32 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
Even less gears is not a doctrinaire puzzle.
So I think this means it can't be expressed using Andreas' notation. Does this also mean that GAP can't be used to calculate it's number of states? Or does anyone see a trick that could be used to model this in GAP?

My mind is telling me you might be able to model this as a doctrinaire puzzle. Think of an Even Less Gears puzzle minus the stickers. There are a finite number of different states reachable where the geared edges are in different positions. I think this number of states can be divided by 24 as many of them should only differ by a global rotation of the entire puzzle. Is it possible to pick a single one of these states and define that as the doctrinaire state? And then to count all the different possible moves that can be made (that reach new unique states i.e. not a global rotation of the doctrinaire state) before this doctrinaire state must be restored? If so AND this set of moves is finite (which I think it must be) than can't we simply count the states of this doctrinaire puzzle and then multiply this by the number of states which are reachable using this finite move list that takes one into the non-doctrinaire space? Not sure my logic is sound here but it strikes me as something similiar to one of the approaces used to analyze the Alternating Cube. There our finite list of moves was all possible 2 move sequences. If this works... and we also consider the Alternating Cube as doctrinaire... then why wouldn't this puzzle also be consider doctrinaire? I would think if you exclude this one you might also have to exclude the Alternating Cube.

Please feel free to point out the holes in my logic. I'm probably way out in left field.
Carl

[EDIT]I've been thinking about this some more. Even if my logic is sound, I think I'm on a very slippery slope. One I think could use this argument (or one very similar) to claim that all bandaged puzzles were doctrinaire.[/EDIT]

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 Post subject: Re: Very Strange SubgroupsPosted: Sat Sep 28, 2013 7:06 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
Bram wrote:
Even less gears is not a doctrinaire puzzle.
I second this. It is the short explanation of what I tried to write here:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2474
BTW: The key tells you that this item has been in the museum for more than one year.

Carl: What you suggest is similar to what Jaap uses for the Mefferts Bandaged Cube:
http://www.jaapsch.net/puzzles/bandage.htm
That one has 440 "configurations" (aka signatures) and 2520 permutations within each signature.

As general approach it is okay. In detail, creating all the sequences starting and ending at the solved configuration, takes a lot of time.

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 Post subject: Re: Very Strange SubgroupsPosted: Sat Sep 28, 2013 12:26 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Andreas Nortmann wrote:
Bram wrote:
Even less gears is not a doctrinaire puzzle.
I second this. It is the short explanation of what I tried to write here:
http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=2474
BTW: The key tells you that this item has been in the museum for more than one year.
Nice description. And I agree. I should have known you had this understood. Why didn't anyone correct me when I was calling Item 2 and Item 6 above the same puzzle as the Even Less Gears and Anisotropic Cube puzzles respectively over a year ago? I see my error now. I was just looking at how the faces could turn from the solved state. I didn't even notice that the different types of edges were ending up in different positions.

Oh... and you left the "f" off of "four" in this sentence in the description.
Quote:
To achieve this he replaced our of the geared edges...
Being one of the poorer spellers on the site I'm not being critical... just trying to help.
Andreas Nortmann wrote:
Carl: What you suggest is similar to what Jaap uses for the Mefferts Bandaged Cube:
http://www.jaapsch.net/puzzles/bandage.htm
That one has 440 "configurations" (aka signatures) and 2520 permutations within each signature.
Nice... glad to see my logic was sound.
Andreas Nortmann wrote:
As general approach it is okay. In detail, creating all the sequences starting and ending at the solved configuration, takes a lot of time.
Oh I never said it was easy or trivial. Notice I didn't even try to do the math yesterday when I was posting about the idea.

However... this really causes me to have issues with calling the Alternating Cube doctrinaire. That puzzle has 2 signatures. The original definition of doctrinaire only allowed 1 signature... the doctrinaire one. If we allow doctrinaire puzzles to have 2 signatures... why not allow 440 as with the Bandaged Cube? Or how many signatures the Even Less Gears puzzle has... which I suspect is well less then 440? Let's see... if the 4 gears can be assigned to any of the 12 edges that is 12 choose 4 or 495 possibilities. But many of these are global rotations of others. But 495/24 isn't a whole number... ok I see why. Lots of the positions are invariant under some of the rotations (the solved state being a good example of one). So I guess that places the minimum number of signatures at 21 (likely a bit higher) assuming there isn't some parity restriction making things lower.

Carl

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 Post subject: Re: Very Strange SubgroupsPosted: Sat Sep 28, 2013 11:19 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Ok... updated the table for the Gear Cube Kit. Feel free to correct me if I've errored on the use of the notation but I think I have it correct.

Thanks,
Carl

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 Post subject: Re: Very Strange SubgroupsPosted: Sun Sep 29, 2013 2:12 am

Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany
wwwmwww wrote:
However... this really causes me to have issues with calling the Alternating Cube doctrinaire. That puzzle has 2 signatures. The original definition of doctrinaire only allowed 1 signature... the doctrinaire one. If we allow doctrinaire puzzles to have 2 signatures... why not allow 440 as with the Bandaged Cube? Or how many signatures the Even Less Gears puzzle has... which I suspect is well less then 440? Let's see... if the 4 gears can be assigned to any of the 12 edges that is 12 choose 4 or 495 possibilities. But many of these are global rotations of others. But 495/24 isn't a whole number... ok I see why. Lots of the positions are invariant under some of the rotations (the solved state being a good example of one). So I guess that places the minimum number of signatures at 21 (likely a bit higher) assuming there isn't some parity restriction making things lower.

I had to look up the Altrnating Cube again. Yes it is a complicated case.
Alternating Cube could be "made doctrinaire" by defining it like (UL' UR' UF' UB' DL' ...)
It is like enumerating the sequences I mentioned above.
It all depends on the precise (!) definition. I have heard voices who do not like slice moves at all. Those voices would never accept the Siamese Cube to be doctrinaire puzzle.

4 gears distributed over 12 edges lead to 27 (rotations only) or 18 (rotations plus reflections) distinct patterns. (Burnsides Lemma)

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 Post subject: Re: Very Strange SubgroupsPosted: Sun Sep 29, 2013 9:38 am

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Andreas Nortmann wrote:
I had to look up the Alternating Cube again. Yes it is a complicated case.
Alternating Cube could be "made doctrinaire" by defining it like (UL' UR' UF' UB' DL' ...)

It's more elegant to express the Alternating Cube using only three permutations, each of which includes mirror imaging the entire puzzle.

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 Post subject: Re: Very Strange SubgroupsPosted: Sun Sep 29, 2013 11:33 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Bram wrote:
Andreas Nortmann wrote:
I had to look up the Alternating Cube again. Yes it is a complicated case.
Alternating Cube could be "made doctrinaire" by defining it like (UL' UR' UF' UB' DL' ...)
It's more elegant to express the Alternating Cube using only three permutations, each of which includes mirror imaging the entire puzzle.
More elegant yes... but it no longer accurately represents the the mechanism available in the physical Alternating Cube as the mirroring operation isn't available to you. Granted gelatinbrain could make this puzzle possible with the mirroring included and I would have to agree that version would be doctrinaire. But on the physical puzzle as presented by Oskar I feel that the list of generating operations as listed by Andreas is more accurate and yes this allows the puzzle to be mapped to a doctrinaire puzzle. However look at the work Jaap has done for the Mefferts Bandaged Cube:
http://www.jaapsch.net/puzzles/bandage.htm
This leads me to believe a set of generating operations could be created (maybe it already has been) that when applied to the Bandaged Cube always return it to the same signature. That puzzle under that list of operations is then by definition doctrinaire. Does that make it fair to call the Bandaged Cube doctriniare? I don't think so or it defeats the purpose of defining the term in the first place. And I would argue that both you and Andreas have provided a valid list of operations that would map the Alternating Cube to a doctriniare puzzle but that doesn't mean the Alternating Cube IS doctrinaire. The only advantage I see of you list over Andreas' is that I believe your list would produce a doctrinaire puzzle with the same number of permutations. Andreas' list would produce a doctrinaire puzzle with half the number of permutations of an actual Alternating Cube. Isn't this the same as if I had the correct list of generating operations for the Bandaged Cube which mapped it to a doctrinaire puzzle of only 2520 permutations, or 1/440th the number of permutations of an actual Bandaged Cube. No one want's to call the Bandaged Cube a doctrinaire puzzle... so I now have serious reservations about calling the Alternating Cube doctrinaire. Sure... its closer to being doctrinaire but if one starts getting into the fuzzy ground of trying to define how close is close enough than it feels like it its counter productive.

Opinions? I'm particularly interested in your's Bram as you first defined the term.

Not trying to argue... just trying to point out that opening up the definition of doctrinaire to include alternating puzzles I think we've taken a step off on a very slippery slope. Let me quote this post from the Alternating Cube thread:
Bram wrote:
The question of whether alternating puzzles are doctrinaire has come up before - both the Spider Gear and Tracker Ball are alternating (okay, people didn't really get into discussion of that with the Tracker Ball, I think because hardly anybody understands it.) The general consensus seems to be that they are in fact doctrinaire because (a) their underlying group can be expressed using permutations, albeit with a mirror imaging trick like I explained before, and (b) they're vertex transitive (okay maybe that's just another way of reiterating the same point.)
And here I think I need to point out that I believe we've been using the term "permutations" to mean different things. I think Bram is stating that the underlying group can be expressed using what I've been calling a list of generating operations. I typically use permutations to refer to the number of states a puzzle can reach. Assuming my understanding is correct I think both uses are valid but I think here this could lead to some confusion if its not pointed out. That said... can't I express the Bandaged Cube as having an underlying group that can be expressed using "permutations", albeit with a trick which would entail a very long list of "permutations". I'm not 100% certain I yet have a grasp on exactly what the term vertex transitive means but I agree with Bram that I think its just another way of saying what is said in point (a). If that is correct, and we allow this consensus to apply to the definition then I believe the logical conclusion would be that the Bandaged Cube is doctrinaire.

That stated I feel we must reject this consensus.

Carl

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