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Jared

Post subject: Generalized Pyraminx and FTO theory? Posted: Thu Aug 23, 2012 9:59 am 

Joined: Mon Aug 18, 2008 10:16 pm Location: Somewhere Else

About a year ago I made this topic, which never really got off the ground. For large cubes, there are 2 types: Even (no fixed center piece) Odd (fixed center piece) However, for large Pyraminxes and FTOs, you can classify them based on the center into 3 groups: No Center (selfexplanatory) Regular Center (the center triangle points in the same direction as the whole face) Inverted Center (the center triangle points in the opposite direction as the whole face) Starting from the Master Pyraminx or FTO (4 layers) because it's the first with a center, that means that we don't hit repetition until the 7layer version. But does that mean that all three solve types are unique in some way (like how even cubes are distinct from odd ones because they can have parity problems)? And, once we do hit repetition (7, 8, and 9 layer versions), is it just "more of the same" like cubes, or are there other factors? (Brandon, I'd love to read what you were going to write in that other topic, but didn't... )


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Brandon Enright

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Thu Aug 23, 2012 10:31 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Hey Jared, I didn't write about this in part because I couldn't figure out the exact properties of each puzzle as you go bigger and bigger. I'm hindered in part because Geltinbrain's 5.1.X series doesn't go very high. I have one of Timur's Royal Pyraminxs now which should help some. Even identifying all of the piece types is somewhat hard. As you have pointed out there is some variety to the center triangle types and there is some variety to the edge pieces too. For Pyraminxs I have a really hard time doing it in my head. Being able to turn the puzzle really helps me work through each pieces behavior. You don't ever run into any parity issues with any sized Pyraminx because every turn is modulo 3 so the whole puzzle stays in an even permutation at all times. I will try again to take a stab at this
_________________ Prior to using my real name I posted under the account named bmenrigh.


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Jared

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Thu Aug 23, 2012 11:42 am 

Joined: Mon Aug 18, 2008 10:16 pm Location: Somewhere Else

Have you tried Ultimate Magic Cube?
Edit: The 6x6 and 7x7 cubes DO add another piece type (centers which are neither + or X), but that's mostly only important in supersolves. Perhaps the 7, 8, and 9 layer Pyraminxes and FTOs also have this property, and 10 is where they start getting redundant...


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Brandon Enright

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Sun Sep 02, 2012 5:57 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Alright I figured this out. So we're talking about Npyraminx where N is the number of layers. The original Pyraminx is N = 3. The Master Pyraminx is N = 4, and so on. I'm not going to bother with N = 1 or 2. Broadly speaking a Npyraminx has the following part types and counts: 4 trivial tips (3 orientations each)
 4 tip bases (3 orientations each)
 6 middleedges (2 orientations each)
 Some number * 12 of edge wings (no orientation)
 0 or 4 centers (3 orientations each but orientation not visible)
 Some number * 12 of wingcenters (no orientation)
So lets tackle the easy parts first. The trivial tips and tip pases are fixed at 4 each. There are two types of edges. "outer edges" and "inner edges". Outeredges are easy to make a pure [1,1] commutator 3cycle for and inneredges can be cycled along with an outeredge with [3,1] commutator 3cycles. If the N for the Npyraminx is odd then the middleedge will be an outeredge and if it is even it'll be a inneredge. The difference between the edge types is superficial. There are always 6 middleedges and the number of edgewings is (N  3) * 12. There are either 0 or 4 centers. If N % 3 == 0 then there are no centers and otherwise there is 1 center for each face for a total of 4. For the number of wingcenters, the number is simple to calculate but figuring it out can be tricky. Each centerwing appears on a face 3 times so you could try to count out how many there are and find a formula. I have highlighted them on the Npyraminx where N in {5, ... 10}: Attachment:
510_cent_unique.png [ 57.29 KiB  Viewed 1232 times ]
If you made a table it would be: 3: 0 4: 0 5: 1 6: 3 7: 5 8: 8 9: 12 10: 16 11: 21 [...] The trick is not to try to find the pattern of highlighted pieces to derive the formula. Instead observe that the total number of pieces in the cent for a face is (N  3) ^ 2. A piece in the center either appears 3 times because it is a centerwing or it appears once because it is a center. That means the number of centerwings in the (total number of center pieces  the center if it's there) / 3. So for a 7pyraminx there are ((7  3)^2  1) / 3 = 5 centerwings. For a 9pyraminx there are ((9  3) ^ 2  0) / 3 = 12 centerwings. So if you will permit me the modulus operator (%) and 0^0 == 1 then... The generalized Npyraminx distinct position formula is:npyra(n) = ((4! / 2) * (3^4)^2 * (6! / 2) * (2^6 / 2) * (12! / 2)^(n  3) * (((4! / 2) * (1  (0^(n % 3)))) + (0^(n % 3))) * (12! / ((3!)^4))^(((n  3)^2  (1  (0^(n % 3)))) / 3)) / 12Which produces: ? npyra(3) % = 75582720 ? npyra(4) % = 217225462874112000 ? npyra(5) % = 19228688422470957567836160000000 ? npyra(6) % = 52425105093745505455227317179687895040000000000000 ? npyra(7) % = 20582183665033346731252760185588627707250626464317440000000000000000000 [...] npyra(17) = 1.48 * 10^488 [...] npyra(100) = 1.43 * 10^18282 As for solving, you can always solve centersout with a combination of intuition, [1,1], and [3,1] commutators. Once you've solved the centers, reduce edges by first grouping all of the inneredgewings to the middleedge and then all of the outeredge wings. Then solve the very shallow reduced HM Pyramid. You could end up with a 3cycle in the centergroups at the end but the fix is simple.
_________________ Prior to using my real name I posted under the account named bmenrigh.
Last edited by Brandon Enright on Tue May 21, 2013 10:51 am, edited 1 time in total.


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Jared

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Sun Sep 02, 2012 10:49 pm 

Joined: Mon Aug 18, 2008 10:16 pm Location: Somewhere Else

Wow! Thanks so much! I was only hoping for a bit on solving, but you went way past that. So, unlike cubes, when it comes to solving, there really isn't a general difference between even/odd or k/k+1/k+2... what a bummer! So, now I'm curious as to how this could be related to FTOs. If you bandage an order3 FTO right, you get an Octaminx, which is a truncated Pyraminx. In fact, the classic FTO is simply 2 Octaminxes in one puzzle, as can be seen by the circle version (GB 4.1.11)  the circles on each of the two separate sets of faces show the underlying Octaminxes, so you have to solve both of them in addition to the FTO itself. So, how do higherorder FTOs relate to higherorder Pyraminxes, if they do?


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Brandon Enright

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Sun Sep 02, 2012 10:52 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Jared wrote: So, how do higherorder FTOs relate to higherorder Pyraminxes, if they do? I dunno, I haven't really thought about it. Obviously the face patterns are roughly the same but on the FTO there are no inneredges, those are two discrete center triangle pieces. Also, the FTOs have center pieces in two orbits. FTOs can probably be thought of as (mostly) two disjoint pyraminxes. I might take a stab at a generalized FTO analysis since it should follow from most of my analysis of the Pyraminx family.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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Jared

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Sun Sep 02, 2012 10:55 pm 

Joined: Mon Aug 18, 2008 10:16 pm Location: Somewhere Else

I really owe you a drink or something for all this.


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Brandon Enright

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Mon Sep 03, 2012 5:17 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Alright I did the big FTOs too. Observe the centers as they get bigger: Attachment:
36_cent_highlight.png [ 32.81 KiB  Viewed 1135 times ]
Note that the nonouterlayer centers grow at the same rate as the Pyraminx analysis above. Also there are always 3 centers on the outside and the number of triangle centers on the edge grow linearly with N. For the edges, there is no difference between middle edges and edge wings. Edgewings are actually two different sets of pieces (in different orbits). They don't have any orientation. There are 12 orientations available to the puzzle rather than 24 due to twist restrictions on the corners and edges. The center pieces are in two orbits. That is, 4 faces are in one orbit and 4 faces are in the other. This is basically like two Pyraminx puzzles that have been fused. Taking all of that into account... The generalized NFTO formula is:nfto(n) = ((6! / 2) * (2^6 / 2) * (((4! / 2)^2 * (1  (0^(n % 3)))) + (0^(n % 3))) * ((12! / ((3!)^4))^2)^((((n  3)^2  (1  (0^(n % 3)))) / 3) + 1 + (n  3)) * (12! / 2)^(n  2)) / 12Here are some interesting values:nfto(3) = 31408133379194880000000 nfto(4) = 147970626354794338756025552732160000000000000 nfto(5) = 661317980389074634392982723584033456560017419062476800000000000000000000000 nfto(6) = 2.804 * 10^114 nfto(7) = 2.465 * 10^158 [...] nfto(17) = 4.649 * 10^1021 [...] nfto(100) = 2.530 * 10^36838 Solvingwise, first reduce the centers using [3,1] commutators. Be sure to reduce the center groups into an even permutation. Then form edgegroups with [3,1] commutators (paring the edge wings up to each other or to a middle edge if there is one). Then complete the edge groups by cycling the triangles between the edge wings with [3,1] commutators. Then solved the reduced, shallowcut FTO. The [3,1] commutators needed for the various piece types are trivial to extend into bigger and bigger NFTOs. The "hardest" NFTOs are the ones where N is divisible by 3 because they don't have the middle center pieces so it's possible to reduce the centers into an odd permutation.
_________________ Prior to using my real name I posted under the account named bmenrigh.
Last edited by Brandon Enright on Tue May 21, 2013 10:52 am, edited 1 time in total.


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Jared

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Mon Sep 03, 2012 6:31 pm 

Joined: Mon Aug 18, 2008 10:16 pm Location: Somewhere Else

Thanks!!!
So, by "odd permutation" you mean a situation kind of like a 4x4's centers being out of order, right? So, if you reduce the centers into the right spots, you can avoid this? Or do you mean that the order6 FTO (for example) can have parity problems towards the end?


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Brandon Enright

Post subject: Re: Generalized Pyraminx and FTO theory? Posted: Mon Sep 03, 2012 6:35 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Jared wrote: Thanks!!!
So, by "odd permutation" you mean a situation kind of like a 4x4's centers being out of order, right? So, if you reduce the centers into the right spots, you can avoid this? Or do you mean that the order6 FTO (for example) can have parity problems towards the end? I mean you could end up with two center groups swapped. You want to make sure you reduce them to positions where this won't happen. The 4x4x4 centers example is not so great because once you reduce to 3x3x3 centers you can't permute the centers relative to each other at all.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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