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 Post subject: Alternating Skewb by OSKAR
PostPosted: Sat Nov 09, 2013 2:15 pm 
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Hi Twisty Puzzle fans,

Alternating Skewb was requested by Bram Cohen at the Twisty Puzzles Forum after discussing Alternating Cube. It is a Skewb in which turns must alternate between clockwise and counter-clockwise. A geared flip-flop mechanism inside assures that every clockwise turn is followed by a counter-clockwise turn and vice verse.

Watch the YouTube video.
Buy at my Shapeways Shop.
Read more at the Shapeways Forum.
Check out the photos below.

Enjoy!

Oskar
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Last edited by Oskar on Sat Nov 09, 2013 2:21 pm, edited 1 time in total.

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Sat Nov 09, 2013 2:21 pm 
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Yet another ingenious mechanism!
I want to ask: did this mechanism inspire your new alternating cube mechanism? Or did you design that first?

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Sat Nov 09, 2013 2:27 pm 
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benpuzzles wrote:
Yet another ingenious mechanism!
I want to ask: did this mechanism inspire your new alternating cube mechanism? Or did you design that first?
Ben,

Thank you. It is the same essentially mechanism. I cadded Alternating Skewb before the new version of Alternating Cube.

Oskar

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Sat Nov 09, 2013 10:34 pm 
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Very well done! If someone gives you a reasonable (and sometimes even an unreasonable) idea for a puzzle, you make it in very little time!

-d


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Mon Nov 11, 2013 1:26 am 
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I wish I had a GAP-file for the Skewb.
I would like to know how restricted this one is.


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Mon Nov 11, 2013 3:58 am 
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Alright I took a stab at this one. I modeled the group generated by all the ways to make a clockwise move followed by a counter-clockwise move on a skewb. There are 12 unique, non-trivial generators that each have an order of 30, which is extremely large by twisty standards. It is also a bit unusual because the cycles performed by a "generator" are not all the same size. Looking at the stickers we have a cycle of 5, two cycles of 3, and two cycles of 6 on every generator. This group is equivalent to the Alternating Skewb where configurations for which a counter-clockwise move must occur are not legal stopping positions, so this group has half the number of configurations as the Alternating Skewb (unless of course you wish to define it this way instead :lol: )

After running some simple custom code that I've been developing for another project, I found that this group (puzzle) has 174,960 reachable elements (configurations). There are y configurations x moves away from solved as given in the following table:
x-y
0-1
1-12
2-108
3-966
4-7296
5-43739
6-102224
7-20535
8-79
Just for fun, 64 of the 79 hardest positions have the centers solved exaclty

Therefore the Alternating Skewb has 349,920 reachable configurations, half that require a clockwise move next and half that require a counter-clockwise move next. These configurations are distributed from solved as:

0-1
1-? (4)
2-12
3-? (36)
4-108
5-?
6-966
7-?
8-7296
9-?
10-43739
11-?
12-102224
13-?
14-20535
15-?
16-79
17-? (possibly 0)

Oddly enough, I don't think it's trivial to fill in the question marks even though I have the exact value at all even depths. The first two are obviously 4 and 36 but I'm not sure about the others... :?

The normal Skewb has 3,149,280 configurations which means the Alternating Skewb has exactly 1/9th the state space of the normal Skewb.

I think that's enough for now. Someone else can contribute more :wink:

Peace,
Matt Galla

PS: Another point of interest that I just noticed: The Skewb can be solved in 11 moves at worst. It's kind of neat that the Alternating Skewb takes several extra moves for some states even though the set of states the Alternating Skewb can reach is strictly contained by the set of states the Skewb can reach 8-)


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Mon Nov 11, 2013 8:35 pm 
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Regarding your end of the video question: Was I correct in saying that you capped the mechanism so it couldn't be manually tampered with?

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Tue Nov 12, 2013 4:22 pm 
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TheCubingKyle wrote:
... you capped the mechanism so it couldn't be manually tampered with?
Correct!

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Fri Nov 15, 2013 10:16 am 
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Looking at # of skewb states of # of alternating skewb states:

3149280 / 349920 = 9

Anybody know where the 9 comes from? That seems to imply that it's from the corner orientations, which would make that part of the solution a bit interesting.


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Thu Dec 12, 2013 5:58 am 
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As I stated in the Alternating Cube thread, I have found a way of mapping the moves and states of the Alternating Skewb to a group, which proves the states of the Alternating Skewb do form a group. I can now easily complete the table:

0 - 1
1 - 4
2 - 12
3 - 36
4 - 108
5 - 324
6 - 966
7 - 2712
8 - 7296
9 - 18908
10 - 43739
11 - 81580
12 - 102224
13 - 69860
14 - 20535
15 - 1536
16 - 79

As you can see, the values of the even distances match my previous table, so my earlier analysis, while incomplete, was accurate. As I stated in the Alternating Cube thread, I can give any of these states (such as the hardest positions to solve) explicitly if someone can suggest a format.

Bram wrote:
Looking at # of skewb states of # of alternating skewb states:

3149280 / 349920 = 9

Anybody know where the 9 comes from? That seems to imply that it's from the corner orientations, which would make that part of the solution a bit interesting.

I looked into this and the answer is not what you may think... Let me demonstrate what I mean:

Perform this sequence, which is valid for the Alternating Skewb, on a solved Skewb:
UBL DFL' UBL DFL' UFR DBR' UBL DBR' UBL DFL' UBL DBR'
Surprised? :wink:

Here's another, again valid for the Alternating Skewb:
UFR DFL' UFR DBR' UBL DFL' UBL UFR' DFL UFR' UBL DBR' DFL DBR'

Clearly something sneaky is going on here... Where does the factor of 9 come from?! Don't worry, I found the answer :)

I want to compare to a normal Skewb, so bear with me for just a second here... On a normal Skewb, the corners come in two orbits. If we leave the physical core of the puzzle fixed, this also fixes the permutation of one orbit of the corners. If we calculate the number of configurations starting here, then accounting for the centers, then the other orbit of corners we reach the following numbers:

a) Each fixed corner can be in 1 of 3 orientations (3^4 = 81)
b) All even permutations of the centers are reachable (6!/2 = 360)
c) The permutation of the remaining corners is isomorphic to the Klein-4 group and can only be in 4 states (4)
d) The sum of the rotations of the remaining corners must be 0 mod 3 (3^4/3 = 27)

This leads to (3^4)(6!/2)(4)(3^3) = 3149280 configurations for the Skewb.

The algorithms for the Alternating Skewb I gave above prove that the factors from the corresponding c) and d) of the Alternating Skewb are the same as the Skewb - a fact which in my opinion is quite surprising! This also means the factor of 9 must come from parts a) and b) alone, and so it does.

a) For the Alternating Skewb, even ignoring the REST of the puzzle, not all 3^4 orientations of the 4 fixed corners are possible! By alternating CW and CCW, the sum of the rotations of these corners can be 0 mod 3. Furthermore, by adding a single CW move on at the end, we can also achieve a rotational sum of 1 mod 3 (say positive is CW). However, it is not possible to achieve a rotational sum of 2 mod 3! Isolating these four corners from the rest of the puzzle, this is actually fairly obvious, but I admit I missed it, and I bet many other great minds on the forum missed it too. The orientation of the last fixed corner depends on the orientation of the previous 3 fixed corners. Of the 3 possibilities for this last corner, only 2 are possible. (3*3*3*2 = 3^3 *2 = 54)

b) Although it is possible on a Skewb, it is impossible to cycle 3 faces on the Alternating Skewb independent of the orientation of the fixed corners. What exactly IS possible? That question isn't easy to answer... Here are some facts:
-The first center can be in any of the 6 faces independent of the orientation of the fixed corners.
-The second center can be in any of the remaining 5 faces.
-The third center has four faces to choose from, but it actually can only be in 2 of those.
-At this point, the fourth, fifth, and sixth centers have no choice - the centers' permutation has been fully determined
Now, you may be inclined to ask: for the third center, which two spots? That depends on both the selected orientations of the fixed corners and the locations of the previous two centers. Suppose the orientations of all four fixed corners are correct and two adjacent centers are solved. Then the third center could obviously be in the correct spot as well, implying that in fact all 6 centers are solved, or the 4 remaining centers could be like this:

UBL UFR' UBL DBR' DFL UBL' DFL DBR' UFR DFL' DBR UBL' DBR DFL'

If instead, the two solved centers are on opposite faces, then the remaining centers could either be solved or like this:

UBL UFR' UBL UFR' DFL UBL' DBR UBL' UFR UBL' DBR DFL' UFR DBR' UBL DBR'

Now it may seem like the first of these two configurations should be able to generate another 2, but that actually invokes a mirror symmetry which would require a switch between CW and CCW, forcing two CW turns in a row. If we use an odd number of moves to attempt to create the equivalent center configuration except with a CCW move next, we can only form the mirror image of the center permutation seen before (ignoring the messed up corners that are a required side-affect of the extra CW move) as you can see here:

UBL UFR' UBL DBR' DFL UBL' DFL DBR' UFR DFL' DBR UBL' DBR (For the experts, this sequence is LITERALLY the one I just gave above with the last move removed. Haha, gotcha!)

Any way you look at it, only 6*5*2 = 60 of the expected 6!/2 = 360 permutations of centers are actually reachable after the orientations of the fixed corners have been set


In summary, for the Alternating Cube:

a) Three fixed corners can be in any of the three orientations, but the final corner can only be in two orientations (3*3*3*2 = 54) (2/3 the possibilities of a Skewb)
b) Only 60 permutations of the centers are reachable (6*5*2 = 30) (1/6 the possibilities of a Skewb)
c) The permutation of the remaining corners is isomorphic to the Klein-4 group and can only be in 4 states (4) (same as Skewb)
d) The sum of the rotations of the remaining corners must be 0 mod 3 (3^4/3 = 27) (same as Skewb)

2/3 * 1/6 = 1/9 and indeed the Alternating Skewb has (3*3*3*2)(6*5*2)(4)(3^3) = 349920 configurations, or 1/9 that of a standard Skewb.


These are fun! Gimme another please! 8-)

Peace,
Matt Galla

PS: As most people have probably guessed, every Alternating Skewb algorithm in this post is optimal, so yes, one of the positions I addressed is a hardest position, as far from solved as possible! :wink:


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Thu Dec 12, 2013 5:38 pm 
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So, is an Alternating Megaminx possible, or does geometry get in the way?


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Fri Dec 13, 2013 10:43 am 
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Location: Marin, CA
Interesting Matt, what happens if you color the faces to have orientation?

Also, what about the alternating pentultimate? Or the alternating little chop? :-)


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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Fri Dec 13, 2013 11:45 am 
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Bram wrote:
Interesting Matt, what happens if you color the faces to have orientation?
Good Question.
Bram wrote:
Also, what about the alternating pentultimate? Or the alternating little chop? :-)
The Alternating Skewb comes closer to having the same number of states as a normal Skewb then the Alternating Cube does when compared to a normal 2x2x2. I believe this is due to the Skewb having 4 axes and the Cube only having 3. I suspect that at some point, as axes are added, that the puzzle will have enough freedom that it will to be able to reach all the states of the non-alternating version. Is this true for the alternating pentultimate or alternating little chop? I don't know... but I suspect that it may be. I'd love to be proven wrong... or to see a proof of the number of axes needed to assure you are able to reach the same solution space.

Hmmm... How would the rules of an alternating puzzle work on a puzzle which jumbles? I think an alternating puzzle may require that the base puzzle be doctrinaire.

Carl

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Fri Dec 13, 2013 12:16 pm 
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I've seen this mechanism before:

Image

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 Post subject: Re: Alternating Skewb by OSKAR
PostPosted: Fri Dec 13, 2013 10:25 pm 
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Jared wrote:
So, is an Alternating Megaminx possible, or does geometry get in the way?

Jaap actually addressed this in the Alternating Cube thread
jaap wrote:
kastellorizo wrote:
Which puzzles give the same number of positions, if we apply to them the clockwise/anticlockwise mechanism?
The megaminx is one, I believe.
If you take two adjacent faces, say A and B, and choose a third face C that is independent from them, then you can do A+ C- B+ A- C+ B-. This is equivalent to A+ B+ A- B- = [A,B]. So you can do the commutator of any two adjacent faces. As the puzzle has no odd permutations, that should enough to be able to solve any position.

And he is correct, a simple commutator is a 3 cycle of edges, a commutator of commutators (a similar argument shows an inverse commutator is possible as well) is a pure 3 cycle of corners, and mixing orientations of these will give pairs reorientations of both corners and edges so the Alternating Megaminx would have the same number of positions as a standard Megaminx. And it would be a pain, but the above procedure is sufficient for solving it as well. The only thing I'm not sure of is if God's Number for the Alternating Megaminx is the same as God's Number for the standard Megaminx; it is very likely larger. Of course, this all changes if we make center orientations visible, but not by much (the last center can only reach 2 of 5 possible orientations). In either case, the groups are much too large to simulate completely :?

Bram wrote:
Interesting Matt, what happens if you color the faces to have orientation?
Thanks! I think so too, and thank you for replying! On a standard Skewb, 5 of the centers can be in 2 orientations while the final center's orientation depends on the previous 5. Sadly, the Alternating Skewb contains no surprises here: it adds exactly the same factor: 2^5 = 32 times as many positions as the puzzle without center orientations.

The updated list of positions at each depth (this one took a bit longer, 36 minutes on my Dell Inspiron 15R laptop):
0 - 1
1 - 4
2 - 12
3 - 36
4 - 108
5 - 324
6 - 966
7 - 2880
8 - 8544
9 - 25236
10 - 73423
11 - 209256
12 - 571600
13 - 1433808
14 - 2865098
15 - 3544768
16 - 2070793
17 - 382328
18 - 8162
19 - 80
20 - 13

Total number of states: 11,197,440

As you can see, the list is the same up until depth 7. This is because two different sequences of 7 moves each produce two configurations that are identical except for some centers reoriented. Running through one sequence forwards and then the other backwards produces the optimal sequence to flip some center orientations without affecting the rest of the puzzle:

UBL UFR' DFL UBL' DFL UBL' DBR UFR' UBL DFL' UFR DFL' UFR DBR'

Of course, the above sequence actually flips 4 centers, to flip only two adjacent centers optimally, you need two more moves:

UBL UFR' UBL UFR' DBR DFL' DBR UBL' DFL DBR' DFL DBR' UFR UBL' UFR DFL'

Bram wrote:
Also, what about the alternating pentultimate?
I suspect Carl is correct here
wwwmwww wrote:
The Alternating Skewb comes closer to having the same number of states as a normal Skewb then the Alternating Cube does when compared to a normal 2x2x2. I believe this is due to the Skewb having 4 axes and the Cube only having 3. I suspect that at some point, as axes are added, that the puzzle will have enough freedom that it will to be able to reach all the states of the non-alternating version.
To prove this, I would have to find sequences in the Alternating Pentultimate that cycle 3 centers, cycle 3 corners, and twist 2 corners. The group is almost certainly too large to model completely, and I suspect optimal solutions to each of these are probably too long as well, so computer trickery won't help me here. Gonna have to resort to good ol' by-hand cleverness. But it would be SOOOO much easier if I could hold a physical Alternating Pentultimate.... :wink:

Bram wrote:
Or the alternating little chop? :-)

If you mean the jumbling Little Chop, then Carl brings up a good question. How does the alternating paradigm work? Of the 5 other states each move can potentially stop at, is only the immediate closest one (either CW or CCW depending on the move) allowed? This means every move would be forced to be jumbling, and that gets very nasty very fast. I'd be surprised if it can be scrambled more than a few turns...

If you mean 180 degree only Little Chop, then c'mon guys, CW and CCW are the same thing! :lol:

Peace,
Matt Galla


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