# FTO thread



## TipsterTrickster (Jul 30, 2020)

I haven't seen any FTO threads yet so I decided to start one.
In case you aren't aware face turning octahedron (aka FTO) is an extremely fun unofficial event that has been taking off recently. Hardware sucks, but the puzzle is a lot of fun to solve, and hopefully better hardware gets made in the near future. There has also been a #FTOforWCA push recently because it is very unique and would fit in very well as an event!


Some links:
cubicle buy link: https://www.thecubicle.com/products/lanlan-octahedron?_pos=1&_sid=86b83f87d&_ss=r
Tutorial: 



Rankings: https://docs.google.com/spreadsheets/d/1Pr7vqSdNtTaGYwQzQMFEQ_NqujVAcZSn8E4ccpOGfJY/edit#gid=0
Weekly comp: http://bit.ly/ftoweekly


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## TipsterTrickster (Jul 30, 2020)

If you are interested in seeing some solves Dan (wr1) and I (wr2) are going head to head tonight at 8pm eastern time US here:


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## CyoobietheCuber (Jul 30, 2020)

How. Did. You. Do. That. BLD?!


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## TipsterTrickster (Jul 30, 2020)

CyoobietheCuber said:


> How. Did. You. Do. That. BLD?!


I haven’t done it blind yet, but I know Ben Streeter and Enoch Gray have, here is enochs video where he describes the method:


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## Micah Morrison (Aug 2, 2020)

I'm pretty new to FTO (PB is 5:08.25) and I'm wondering if lubing a lanlan FTO with Vaseline would be a good idea, since that's often what people recommend for Rubik's brand 3x3s since it eats away at the plastic. I would think the same principle would apply to the lanlan FTO, but I wasn't too sure.


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## zslane (Aug 2, 2020)

Just in case you haven't seen this video already:


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## Micah Morrison (Aug 3, 2020)

zslane said:


> Just in case you haven't seen this video already:


I've seen that, but not sure if I want to commit that much time and work to modding my FTO. I was wondering if lubing it with Vaseline would somewhat substitute for modding it.


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## TipsterTrickster (Aug 3, 2020)

I personally wouldn't recommend lubing with Vaseline, I use weight 5 and it works fine.


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## zslane (Aug 3, 2020)

I'd stay away from Vaseline:


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## Micah Morrison (Aug 3, 2020)

TipsterTrickster said:


> I personally wouldn't recommend lubing with Vaseline, I use weight 5 and it works fine.


ok, good to know, I just thought vaseline might be better since it eats away at the plastic.


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## zslane (Aug 3, 2020)

Heh. Vaseline as an alternative to sanding down the pieces? Interesting idea!


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## PapaSmurf (Sep 10, 2020)

Just started to "speed" solve my (currently unmodified) FTO. I have a couple of questions: is there a random state scrambler/is the scrambler in CSTimer random state, do I scramble with green on F or red on F and is there a set of "algs" that are sledges/hedges for every L3T case?


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## xyzzy (Sep 10, 2020)

PapaSmurf said:


> is there a random state scrambler/is the scrambler in CSTimer random state


No(t yet) and no.



PapaSmurf said:


> do I scramble with green on F or red on F


Same as the other WCA puzzles. White (or the lightest colour) on top, with one corner pointing at you; green (or the darkest possible colour) in front, with one corner pointing up. (Note that U and F are not adjacent faces with Ben's notation. In SEE's current notation, the top face has a corner pointing _away_ and the front face has a corner pointing _down_, and the U and F faces are adjacent.)





(I've never done FTO so I can't answer your last question.)


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## PapaSmurf (Sep 10, 2020)

Ok, thanks for that! I really should learn to code. I should this year at university, then (if someone hasn't beaten me to it) I could code one. 
That's the way I've been scrambling too, so that's good.


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## qwr (Nov 12, 2020)

zslane said:


> Just in case you haven't seen this video already:



I didn't watch the full video but I just got a lanlan FTO today and while turning is ok for a non-WCA puzzle, I took it apart and it's abundantly obvious why the puzzle catches so much. None of the sharp corners inside are rounded at all and in particular the core center triangles catch even when no other pieces are in. I think I might mod my FTO later just by sanding down all sharp edges I see and that would probably get rid of a lot of the catching.


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## Kit Clement (Nov 12, 2020)

qwr said:


> I took it apart and it's abundantly obvious why the puzzle catches so much. None of the sharp corners inside are rounded at all and in particular the core center triangles catch even when no other pieces are in.



Welcome to pre-2011 cubing!


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## qwr (Nov 12, 2020)

Kit Clement said:


> Welcome to pre-2011 cubing!


I was in a group chat with some cubing friends and I made the same comment ("remember when puzzle modding was a thing")
I wonder if there's a market for modded FTOs or 3D printed ones yet. Cube manufacturers like qiyi should honestly make a decent FTO just to get it popular and give credibility in the push for WCA.


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## PapaSmurf (Nov 12, 2020)

I hope they do. It would be pretty cool to have a good one.


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## qwr (Nov 12, 2020)

PapaSmurf said:


> I hope they do. It would be pretty cool to have a good one.


I think the cubicle has 3d printers and probably has the resources to semi mass produce 3d printed designs like a FTO, although it would be noticeably more expensive (maybe selling for $30-40)

I've been thinking there may be a market for semi mass produced twisty puzzles in between the original designer printing at home and cube manufacturers with large scale injection molding. However many puzzles already have decent enough mass produced versions that it wouldn't be worth 3d printing, FTO being an exception.


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## tx789 (Nov 12, 2020)

qwr said:


> I think the cubicle has 3d printers and probably has the resources to semi mass produce 3d printed designs like a FTO, although it would be noticeably more expensive (maybe selling for $30-40)
> 
> I've been thinking there may be a market for semi mass produced twisty puzzles in between the original designer printing at home and cube manufacturers with large scale injection molding. However many puzzles already have decent enough mass produced versions that it wouldn't be worth 3d printing, FTO being an exception.



You need a specific type of 3d printing. And fdm doesn't work for twisty puzzles.


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## OreKehStrah (Nov 12, 2020)

I just got an FTO the other day and was wondering why there aren’t scramblers for it in a more megaminx style scramble? I think it might be easier to do some reduction of R+-, U+-, and L+- moves instead of having to deal with each face. I don’t know much about the puzzle yet, so idk if it’s possible to achieve random state scrambles with such a method but I thought I would throw out the idea since I haven’t seen it mentioned yet.


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## PapaSmurf (Nov 12, 2020)

Equally, hold with a corner in UF and do R±, D± or F±. I'll write a bit of python in a bit to do this.


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## OreKehStrah (Nov 12, 2020)

PapaSmurf said:


> Equally, hold with a corner in UF and do R±, D± or F±. I'll write a bit of python in a bit to do this.


With the current state of hardware, I think RUL might be better since it would probably be easier to turn quickly than RDF. That would be my guess though.


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## PapaSmurf (Nov 12, 2020)

Have some code.


Spoiler: Code, probably inefficient





```
import numpy as np
n=0
b=[]
c=0
a=0
u = np.array(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9'])
m=0
d=0
e=0

while n<30:
    a = np.random.randint(1,3)
    n = n+1
    if a == 1:
        c = '+'
    else:
        c= '-'
    b = np.append(b, c)   

while m<10:
    d = np.random.randint(1,3)
    m = m+1
    if d == 1:
        e = ''
    else:
        e= "'"
    u[(m-1)] = e

print('D' + b[0] + ' R' + b[1] + ' F' + b[2] + ' U' + u[0] + ' D' + b[3]
      + ' R' + b[4] + ' F' + b[5] + ' U' + u[1] + ' D' + b[6] + ' R' + b[7]
      + ' F' + b[8] + ' U' + u[2]  + ' D' + b[9] + ' R' + b[10] + ' F'
      + b[11] + ' U' + u[3] + ' D' + b[12] + ' R' + b[13] + ' F' + b[14]
      + ' U' + u[4] + ' D' + b[15] + ' R' + b[16] + ' F' + b[17] + ' U'
      + u[5] + ' D' + b[18] + ' R'  + b[19] + ' F' + b[20] + ' U' + u[6]
      + ' D' + b[21] + ' R' + b[22] + ' F' + b[23]  + ' U' + u[7] + ' D'
      + b[24] + ' R' + b[25] + ' F' + b[26] + ' U' + u[8] + ' D' + b[27]
      + ' R'  + b[28] + ' F' + b[29] + ' U' + u[9])
```


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## qwr (Nov 13, 2020)

PapaSmurf said:


> Have some code.
> 
> 
> Spoiler: Code, probably inefficient
> ...



it was going pretty well until that last line


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## qwr (Nov 13, 2020)

tx789 said:


> You need a specific type of 3d printing. And fdm doesn't work for twisty puzzles.


why not? i've seen plenty of 3d printed puzzles.


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## abunickabhi (Nov 13, 2020)

Ben has made really high quality videos on the FTO, and it feels good that such good videos resources exist on the internet. Sadly, I live in India and buying a new FTO is a bit of a hassle for me.


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## qwr (Nov 13, 2020)

abunickabhi said:


> Ben has made really high quality videos on the FTO, and it feels good that such good videos resources exist on the internet. Sadly, I live in India and buying a new FTO is a bit of a hassle for me.


can you get cheap/free shipping from chinese stores like cubezz/ziicube/hknowstore?


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## TipsterTrickster (Nov 13, 2020)

A few notes:


qwr said:


> I was in a group chat with some cubing friends and I made the same comment ("remember when puzzle modding was a thing")
> I wonder if there's a market for modded FTOs or 3D printed ones yet. Cube manufacturers like qiyi should honestly make a decent FTO just to get it popular and give credibility in the push for WCA.





qwr said:


> I think the cubicle has 3d printers and probably has the resources to semi mass produce 3d printed designs like a FTO, although it would be noticeably more expensive (maybe selling for $30-40)
> 
> I've been thinking there may be a market for semi mass produced twisty puzzles in between the original designer printing at home and cube manufacturers with large scale injection molding. However many puzzles already have decent enough mass produced versions that it wouldn't be worth 3d printing, FTO being an exception.



Ben Streeter has a 3D printed fto and used it for a while, but retired it because it became too worn down. From what I know it worked better than the lan lan, popped less, and had less corner twists.



tx789 said:


> You need a specific type of 3d printing. And fdm doesn't work for twisty puzzles.



You can use FDM for 3D printed puzzles it just requires more post processing than SLS, and either method would probably take way too much time to be practical, and they would probably end up costing $100+ 



OreKehStrah said:


> I just got an FTO the other day and was wondering why there aren’t scramblers for it in a more megaminx style scramble? I think it might be easier to do some reduction of R+-, U+-, and L+- moves instead of having to deal with each face. I don’t know much about the puzzle yet, so idk if it’s possible to achieve random state scrambles with such a method but I thought I would throw out the idea since I haven’t seen it mentioned yet.



We don’t use megaminx style scrambles because 
a) assigning a letter to each face is easy enough and is fast enough to be practical b) it’s easier to make a random state scrambler, and random move scrambles are more efficient and effective. Also fto definitely needs a random state scrambler whereas megaminx doesn’t really, I haven’t done the math, but using megaminx style scramblers will reach much less states than the scramblers we currently use (megaminx is ~3% iirc) and with the kinda luck fto has I think we should be able to reach all the states, also this is why the fto community recently switched from 25 to 30 move scrambles.


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## qwr (Nov 13, 2020)

TipsterTrickster said:


> either method would probably take way too much time to be practical, and they would probably end up costing $100+


why do you say this? is it the labor in assembling the pieces? because I don't think it would be that much in raw materials.
AFAIK when Chris Tran was at the cubicle they 3d printed a lot of tools and experimented with selling a custom clock.


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## TipsterTrickster (Nov 13, 2020)

qwr said:


> why do you say this? is it the labor in assembling the pieces? because I don't think it would be that much in raw materials.
> AFAIK when Chris Tran was at the cubicle they 3d printed a lot of tools and experimented with selling a custom clock.


With the outer casing for the angstrom clock it takes like an hour to print depending on your settings (when I printed a replacement for mine it took an hour and a half, but that was using a shorter layer height than what they used), when printing a puzzle, due to all the parts it could take 10-15+ hours for 1 puzzle, then on top of that due to the layer lines you need to sand down and/or use acetone to smooth everything out and make it useable, they didn’t have to do this with the clocks outer casing, then finally assemble, sticker and test it, and even after all that it will still need some breaking in. It is definitely possible (as Ben has proved) to make a really good 3D printed fto, however I don’t think it is practical for mass production. Maybe SLS would work better for it, but Idk much about SLS printing and how much post production is required to make those puzzles good.


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## qwr (Nov 13, 2020)

TipsterTrickster said:


> With the outer casing for the angstrom clock it takes like an hour to print depending on your settings (when I printed a replacement for mine it took an hour and a half, but that was using a shorter layer height than what they used), when printing a puzzle, due to all the parts it could take 10-15+ hours for 1 puzzle, then on top of that due to the layer lines you need to sand down and/or use acetone to smooth everything out and make it useable, they didn’t have to do this with the clocks outer casing, then finally assemble, sticker and test it, and even after all that it will still need some breaking in. It is definitely possible (as Ben has proved) to make a really good 3D printed fto, however I don’t think it is practical for mass production. Maybe SLS would work better for it, but Idk much about SLS printing and how much post production is required to make those puzzles good.



I've done a small amount of 3D printing (I believe with Ultimaker 3s) where even with the layer lines the surface is quite smooth requiring virtually no sanding, which brings down the manual labor part a lot. It is possible to fit several pieces onto one metal plate. And usually in the makerspace on campus we had running probably 16 small printers and at least 3 large really big ones, along with SLA printers. Admittedly idk much about 3d printing.

Another cost effective option may be selling modded lanlan FTOs. It is probably extremely labor intensive and not worth it.


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## tx789 (Nov 15, 2020)

qwr said:


> why not? i've seen plenty of 3d printed puzzles.


They use sls most if the time.


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## OreKehStrah (Nov 22, 2020)

I mentioned it in the SS discord, but I figured I would throw it out here where it might be seen a bit more:

Any thoughts on calling the FTO "Octant" ? I think this is a good name for the puzzle as an event.


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## qwr (Nov 22, 2020)

OreKehStrah said:


> I mentioned it in the SS discord, but I figured I would throw it out here where it might be seen a bit more:
> 
> Any thoughts on calling the FTO "Octant" ? I think this is a good name for the puzzle as an event.


why? it has nothing to do with octants, which are a real term (such as in octrees)


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## OreKehStrah (Nov 22, 2020)

qwr said:


> why? it has nothing to do with octants, which are a real term (such as in octrees)


Why not? It sounds good. Oct - 8, Octahedron - 8 sides

Why do we call megaminx megaminx? Why not call it the Hungarian Supernova? 

And you can technically divide up each axis of it to fit within each octant of R3


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## qwr (Nov 22, 2020)

OreKehStrah said:


> Why not? It sounds good. Oct - 8, Octahedron - 8 sides
> 
> Why do we call megaminx megaminx? Why not call it the Hungarian Supernova?
> 
> And you can technically divide up each axis of it to fit within each octant of R3



because octahedron is a much more accurate term.

and yes megaminx is a stupid name but we're stuck with it now.


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## zslane (Nov 22, 2020)

Octahedron makes sense, but apart from the "cube", none of the other puzzles shaped as a platonic solid have the canonical name. We have pyraminx instead of tetrahedron, megaminx instead of dodecahedron, and icosaix instead of icosahedron. Which is fine, I guess. The name seems less important than the quality of the puzzle. "FTO" will probably stick, even if it gets added as an official WCA event.


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## qwr (Nov 22, 2020)

FTO is actually probably the most descriptive name of any potential puzzle to the WCA because it describes the shape and the turning.


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## DNF_Cuber (Nov 22, 2020)

qwr said:


> FTO is actually probably the most descriptive name of any potential puzzle to the WCA because it describes the shape and the turning.


pyraminx being the best current, FMC is in a grey area


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## OreKehStrah (Nov 22, 2020)

There really isn't a need for the name to accurately describe the event though. 
If we use face turning octahedron as the name, pretty much everyone is going to refer to it as FTO. As a result, if someone hears FTO, they will either know what it is or they won't, so the whole the-name-describes-the-event goes out the window for the most part.

However, names can be interesting and still convey some idea of the puzzle without directly describing it, such as square-1 and pyraminx. Square-1 somewhat references bringing the cube back into cubeshape, and pyraminx has a bit of a description of itself as a name.

I think FTO is fine, but why not give it a fun name like Octant? It somewhat gives an idea of the puzzle and feels similar enough to other event names.

anyway, it's not like there's a right or wrong answer, nor does it really matter since it's not an official event. I just thought I would put an idea out before we permanently settle on just a boring set of 3 letters!


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## PapaSmurf (Nov 22, 2020)

I think it's a very interesting set of 3 letters. For one, they aren't in alphabetical order - very interesting. Why does the alphabet even have an order?

But actually, FTO works and hasn't made people who are into it become not into it. Having a name that is universally accepted is much easier than changing it and it won't confuse people who are new to the puzzle as they can find Ben's guides on the internet, instead of searching something completely different that has even fewer resources or is completely different. I agree, the name could have a bit more pazazz, but as it is now doesn't disadvantage the puzzle.


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## qwr (Nov 22, 2020)

if you invented it in 2003 (or 1983? http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1663) you could name it that. but now it's FTO


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## Kit Clement (Nov 23, 2020)

I will now begin calling 3x3x3 the FTC and Megaminx the FTD henceforth. Unsure about whether Pyra should be renamed to FTT or CTT.

I feel like Octaminx makes the most sense based on current terminology, but it uses a suffix that is more associated with being an Uwe creation which doesn't apply here. I'm personally fine with FTO.


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## PapaSmurf (Nov 23, 2020)

Mathematically, it does make sense to call them the FTC, FTD and FTT. Therefore everyone should. Octaminx does also make sense but is the same effort to say and more to type.


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## qwr (Nov 23, 2020)

Kit Clement said:


> Unsure about whether Pyra should be renamed to FTT or CTT.


Since the corner is opposite the face, it could be both, but the mechanism arguably shows it to be "corner turning".

I think 3x3x3 is an unambiguous enough name because afaik there is only one way for a 3x3x3 to turn given the cuts and piece shape. With triangular pieces, there is some ambiguity between which cutting planes are on the puzzle, hence corner-turning / 6 axis octahedron vs face-turning / 8 axis octahedron.

Pyraminx is an okayish name but megaminx is pretty undescriptive. Dodecaminx or Pentaminx would've been better.


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## zslane (Nov 23, 2020)

Megaminx led to a whole family of puzzles with metric prefixes: Kilominx, Gigaminx, Teraminx, etc. So at least there is a discernable naming scheme applied to the dodecahedron genotype. I'm not sure what the -minx suffix is meant to describe, but it is what it is and I seriously doubt a renaming effort would succeed.


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## qwr (Nov 23, 2020)

zslane said:


> Megaminx led to a whole family of puzzles with metric prefixes: Kilominx, Gigaminx, Teraminx, etc. So at least there is a discernable naming scheme applied to the dodecahedron genotype. I'm not sure what the -minx suffix is meant to describe, but it is what it is and I seriously doubt a renaming effort would succeed.



Uwe Meffert had this weird trend of using -minx (I think Cubing Historian had a video on this. Also http://tonyfisherpuzzles.net/025 Octaminx ( Rubik type puzzle ).html) 

I'm pretty sure the metrix prefixes weren't intended by Meffert (mega probably just for its informal meaning). AFAIK the first puzzle in the megaminx family other than the megaminx was the gigaminx and named as a joke? If you search the twisty puzzles forum you can probably the original post.


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## ProStar (Dec 10, 2020)

Bump 

I plan on getting an FTO soon. What's the least awful FTO available?


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## Nir1213 (Dec 10, 2020)

ProStar said:


> Bump
> 
> I plan on getting an FTO soon. What's the least awful FTO available?


to add on to that, fto is going to be added to the weekly competition soon.


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## Kit Clement (Dec 10, 2020)

ProStar said:


> Bump
> 
> I plan on getting an FTO soon. What's the least awful FTO available?



Currently, the LanLan is the best available option. Luckily, it's actually in stock right now at the Cubicle: https://www.thecubicle.com/products/lanlan-octahedron


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## xyzzy (Dec 15, 2020)

FTO in GAP. (Using Ben's notation, so U and F are non-adjacent faces.)



Spoiler: Facelet labelling





```
U									B				
		8	7	6	5	4					67	68	69	70	71		
	44		3	2	1		49			22		64	65	66		35	
	43	39		0		46	50			23	19		63		30	34	
L	42	38	36		45	47	51	R	BR	24	20	18		27	29	33	BL
	41	37		9		48	52			25	21		54		28	32	
	40		10	11	12		53			26		57	56	55		31	
		13	14	15	16	17					62	61	60	59	58		
				F									D
```
(I'm using 0-based indices above, but GAP actually does not support that. 0 is replaced with 100 in the GAP code below.)




```
U := (100,4,8)(1,6,3)(2,5,7)(9,22,35)(45,67,44)(47,68,43)(46,69,39)(50,70,38)(49,71,36);
# F := (9,13,17)(10,15,12)(11,14,16)(100,31,26)(36,58,53)(38,59,52)(37,60,48)(41,61,47)(40,62,45);
mirror := (100,36)(1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71);
rotate := (100,18)(1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71);
t2 := (100,9)(1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71);
F := t2 * U * t2;
L := mirror * U^2 * mirror;
R := t2 * L * t2;
BR := rotate * U * rotate;
BL := rotate * F * rotate;
B := rotate * R * rotate;
D := rotate * L * rotate;

# IdenticalSolutionsGroup and TrueSize from 
# http://twistypuzzles.com/forum/viewtopic.php?p=247974#p247974
IdenticalSolutionsGroup := 
function(group, indistinguishable_sets)
    local domain, g1, g2;
    domain := Set(Flat(OrbitsDomain(group)));
    SubtractSet(domain, Flat(indistinguishable_sets));
    g1 := Stabilizer(group, domain, OnTuples);
    g2 := Stabilizer(g1, indistinguishable_sets, OnTuplesSets);
    return g2;
end;;

TrueSize := 
function(group,indistinguishable_sets)
    return Size(group)/Size(IdenticalSolutionsGroup(group,indistinguishable_sets));
end;;

same := [[2,5,7],[11,14,16],[20,23,25],[29,32,34],[38,41,43],[47,50,52],[56,59,61],[65,68,70]];;

G := Group([U, L, F, R, BR, B, BL, D]);;

# gap> TrueSize(G, same);
# 376897600550338560000000
# gap> Size(G);
# 158260810061489362698240000000
#etc.
```

Some interesting stuff:

| ⟨U, F, BL, BR⟩ | = 933120 is really tiny compared to | ⟨U, F, L, R⟩ | = 63712967786496000000, even though the former move set moves every facelet, while the latter move set doesn't. This is because the former group is actually the same as the group of pyraminx-sans-tips states: we now have blocks formed from a corner and two triangles (equivalent to a pyra edge) and from three edges and three triangles (pyra corner). (Note: The actual number of states generated by {U, F, L, R} is somewhat smaller (12290310144000000) due to the triangles having identical copies.)

While the above figure for the size of G seems like it overcounts the FTO's states, that's because it (sort of) does: all 12 orbit-preserving rotations (rotational tetrahedral symmetry) lie in G, so each state gets counted 12 times over, once for each of the 12 different global orientations. This can be "fixed" by using U, L, F, R, Uw, Lw, Fw, Rw as the generating set instead, so one corner piece is stationary and everything else moves around it. (Fixing an edge piece also works. Fixing a centre piece works on the supercube version of the FTO, but not on the FTO itself, since there are still multiple identical centre pieces.)

We don't need all eight faces to generate every state, for that matter. I think six faces is necessary and sufficient. Not sure if allowing wide moves (or slice moves) changes this.


```
G6 := Group([L,F,R,BR,B,BL]);;
hom6 := EpimorphismFromFreeGroup(G6:names:=["L", "F", "R", "BR", "B", "BL"]);;

# these should print true
U in G6;
D in G6;

PreImagesRepresentative(hom6, U);
PreImagesRepresentative(hom6, D);
```

So we have:


Spoiler: long program output





```
U = 
F^-1*L^-1*F^2*(L*F*L^-1*F^-1)^2*L*F^-1*L^-1*F*L*F*(L^-1*F*L*F^-1)^3*L^-1*F^-1*L*(L*F)^2*L*F^-1*(L*F)^2*L^-1*(F^-1*L^-1\
*F*L*F*L^-1*F*L)^2*F^-1*L*F^-1*(L^-1*F)^2*L*F^-1*L*F*L^-2*F^-1*L*F*L^-1*F*L*F*L^-1*F^-1*(F^-1*L)^2*F*L^-1*R^-1*B^-1*L^\
-1*B*L*R*F^-1*L^-2*F*(L*F*L^-1*F^-1)^2*L^-1*(F^-1*L)^2*F*(L*F^-1)^2*L^-1*F^-1*(F^-1*L^-1*F*L)^3*L*F^-1*R^-1*F*L*F^-1*L\
^-1*F^-1*R*F*L^-1*F*L*F^-1*L^-1*F^-1*R^-1*F*L*F*L^-1*F^-1*R*(F*L^-1*F*L)^2*F*L^-1*F^-1*L*F*L^-1*R^-1*F*L*F^-1*L^-1*F^-\
1*R*L^2*F*L^-1*F^-2*L^-1*F*L*F*R*F^-1*L*F*R^-1*L*F*R*L*R^-2*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R\
*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*R^-1*BR*F^-1*BR^-1*BL^-1*F*BL*F^\
-1*L;

# and

D =
(F^-1*L^-1*F^2*(L*F*L^-1*F^-1)^2*L*F^-1*L^-1*F*L*F*(L^-1*F*L*F^-1)^2*L^-1*F*R^-1*F*L*F*L^-1*F^-1*R*L*F^2*L^-1*F^-1*L*F\
^-1*L^-1*F^-1*L*F*L*(F^-1*L^-1)^2*F*(L^-1*F^-1)^2*L^-2*F*L*F*L^-1*R^-1*F*L*F*L^-1*F^-1*R*(L*F*L^-1*F)^2*(L*F^-1)^2*(L^\
-1*F)^2*L*F^-1*L*F*L^-2*F^-1*L*(F*L^-1)^2*(F^-1*L)^2*F*(L*F^-1)^2*(L^-1*F^-1)^2*L*F^-1*L^-1*F*L*F^-1*L*F*R^-1*L^-1*B^-\
1*L*B*R*L*F^-1*(L^-1*F)^2*R^-1*F*L*F*L^-1*F^-1*R*F*L*F^-1*L^-1*F^2*L*F^-1*L^-1*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^2*L^\
-1*F^-1*L^2*F*L^-1*F^-1*L^-2*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^\
-1*L*F*R*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*B\
R*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*B\
L*L*B^-1*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)\
^3*R*F^-1*L^-1*F*R^-1*L^-1*B^-1*L*B*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*B\
R^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*R^-2*L^-1*F*L*R^2*F*R^-1*L*F^-1*R*B*R^-1*B^-1*(R^-1\
*F^-1)^2*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*L*BL*B*BL^-1*L*BR*R^-1*BR^-1*B^-2*R^-2*F^-1*R*BR*F*BR\
^-1*L^-1*F^-1*BL^-1*F^-1*R*BL*F^-1*BR^-1*B*BR*L^-1*F^-1*R^-1*BL*B^-1*BL^-1*R^-1*BR*B*L^-1*B^-1*F^-1*BR^-1*R^-1*B^-1*R*\
B*R^-1*BR*R*BR^-1*R^-1*BL*BR*B*BL^-1*L^-1*BR^-1*F*L*BL*B^-1*BL^-1*BR^-1*L^-1*F^-1*L^2*F*L^-1*(F^-1*L)^2*R*F^-1*L^-1*F*\
R^-1*L^-1*F^-1*L^-1*B^-1*L*B*F*R^-1*L^-1*F*L*F*L^-1*BL^-1*F^-1*BL*R^-1*BR^-1*B*BR*L^-1*BR*B*L^-2*B^-1*L^-1*BR^-1*R^-1)\
^2*(L^-1*F^-1*L*F)^2*L^-1*F*L*F*L^-1*F^-1*(F^-1*L)^2*F*L^-1*F^-1*L^-1*(F*L)^2*F^-1*L^-3*F*L*F^-1*L*(F^-1*L^-1*F*L^-1)^\
2*F^-1*L*F*L^-1*F*(L*F^-1)^2*L^-1*F^-1*R^-1*F*L*F*L^-1*F^-1*R*F^-1*L^-1*F^-1*(L*F^-1*L^-1*F)^3*L*F^-1*L^-1*R^-1*F*L*F^\
-1*L^-1*F^-1*R*F^-1*L*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^-1*L^-1*F^-1*L*(L*F)^2*L*F^-1*(L*F)^2*L^-1*(F^-1*L^-1*F*\
L*F*L^-1*F*L)^2*F^-1*L*F^-1*(L^-1*F)^2*L*F^-1*L*F*L^-1*(F^-1*L^-1*F*L)^2*F^-1*L*F*L^-1*R^-1*B^-1*L^-1*B*L*R*F^-1*L^-1*\
F*L^-1*F^-1*L*(F*L^-1)^2*(F^-1*L)^2*F*(L*F^-1)^2*(L^-1*F^-1)^2*L*F^-1*L^-1*F*(F*L)^2*(F*L^-1)^2*F^-1*(L^-1*F)^2*(L*F)^\
2*L^-1*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*F^-1*L^2*F^-1*L^-1*F*L*F^-1*L*F*L*F^-1*L^-1*F*L^-2*F*L*F^-1*L^-1*F^-1*L*F^-1*L^-\
1*F*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^2*L^-1*F^-1*L^2*F*L^-1*F^-1*L^-2*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^\
-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*L*F*R*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^\
-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR\
^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*L*B^-1*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R\
^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^3*R*F^-1*L^-1*F*R^-1*L^-1*B^-1*L*B*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1\
*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*R^-2*L^-1*F*L\
*R^2*F*R^-1*L*F^-1*R*B*R^-1*B^-1*(R^-1*F^-1)^2*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*L*BL*B*BL^-1*L*\
BR*R^-1*BR^-1*B^-2*R^-2*F^-1*R*BR*F*BR^-1*L^-1*F^-1*BL^-1*F^-1*R*BL*F^-1*BR^-1*B*BR*L^-1*F^-1*R^-1*BL*B^-1*BL^-1*R^-1*\
BR*B*L^-1*B^-1*F^-1*BR^-1*R^-1*B^-1*R*B*R^-1*BR*R*BR^-1*R^-1*BL*BR*B*BL^-1*L^-1*BR^-1*F*L*BL*B^-1*BL^-1*BR^-1*L*F^-1*L\
^-1*F*L*F*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*L*F*R*B*R^-1*B^-\
1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R\
^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*L*B^-1*L^-1*R^-1*\
BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^2*R*F^-1*(L*F*R^-1\
)^2*F^-1*L^-1*F*(R^-1*L^-1)^2*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R^2*F*R^-1*L*R*B*R^-1*B^-1*R^-1*F^-1*R*BL^-1*F*BL*BR^-1*B\
*BR*L^-1*F^-1*R^-1*BL*B*BL^-1*L*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*R^-1*F^-1*BL^-1*F^-1*R^-1*B^-1*(F^-1\
*L^-1)^2*F^2*(L*F*L^-1*F^-1)^2*L*F^-1*L^-1*F*L*F*(L^-1*F*L*F^-1)^2*L^-1*F*R^-1*F*L*F*L^-1*F^-1*R*L*F^2*L^-1*F^-1*L*F^-\
1*L^-1*F^-1*L*F*L*(F^-1*L^-1)^2*F*(L^-1*F^-1)^2*L^-2*F*L*F*L^-1*R^-1*F*L*F*L^-1*F^-1*R*(L*F*L^-1*F)^2*(L*F^-1)^2*(L^-1\
*F)^2*L*F^-1*L*F*L^-2*F^-1*L*(F*L^-1)^2*(F^-1*L)^2*F*(L*F^-1)^2*(L^-1*F^-1)^2*L*F^-1*L^-1*F*L*F^-1*L*F*R^-1*L^-1*B^-1*\
L*B*R*L*F^-1*(L^-1*F)^2*R^-1*F*L*F*L^-1*F^-1*R*F*L*F^-1*L^-1*F^2*L*F^-1*L^-1*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^2*L^-1\
*F^-1*L^2*F*L^-1*F^-1*L^-2*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1\
*L*F*R*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*\
F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*\
L*B^-1*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^3\
*R*F^-1*L^-1*F*R^-1*L^-1*B^-1*L*B*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^\
-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*R^-2*L^-1*F*L*R^2*F*R^-1*L*F^-1*R*B*R^-1*B^-1*(R^-1*F\
^-1)^2*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*L*BL*B*BL^-1*L*BR*R^-1*BR^-1*B^-2*R^-2*F^-1*R*BR*F*BR^-\
1*L^-1*F^-1*BL^-1*F^-1*R*BL*F^-1*BR^-1*B*BR*L^-1*F^-1*R^-1*BL*B^-1*BL^-1*R^-1*BR*B*L^-1*B^-1*F^-1*BR^-1*R^-1*B^-1*R*B*\
R^-1*BR*R*BR^-1*R^-1*BL*BR*B*BL^-1*L^-1*BR^-1*F*L*BL*B^-1*BL^-1*BR^-1*L^-1*F^-1*L^2*F*L^-1*(F^-1*L)^2*R*F^-1*L^-1*F*R^\
-1*L^-1*F^-1*L^-1*B^-1*L*B*F*R^-1*L^-1*F*L*F*L^-1*BL^-1*F^-1*BL*R^-1*BR^-1*B*BR*L^-1*BR*B*L^-2*B^-1*L^-1*BR^-1*R^-1*(L\
^-1*F*(L*F^-1)^2*L^-1*F^-1*R^-1*F*L*F*L^-1*F^-1*R*F^-1*L^-1*F^-1*(L*F^-1*L^-1*F)^3*L*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1\
*R*F^-1*L*F*L^-1*F^-1*L*F^-1*L^-1*F^-1*L*F*L*(F^-1*L^-1)^2*F*(L^-1*F^-1)^2*L^-2*F*L*F*L^-1*R^-1*F*L*F*L^-1*F^-1*R*L*F*\
L^-1*F*L*F*L^-1*F^-1*(F^-1*L)^2*F*L^-1*R^-1*B^-1*L^-1*B*L*R*F^-1*L^-2*F*L*F*L^-2*(F^-1*L)^2*F*(L*F^-1)^2*(L^-1*F^-1)^2\
*(L*F^-1*L^-1*F)^5*F*R^-1*F*L*F*L^-1*F^-1*R*F*L^2*F^-1*L^-1*F*L*F^-1*L*F*L*F^-1*L^-1*F*L^-2*F*L*F^-1*L^-1*F^-1*R^-1*F*\
L*F^-1*L^-1*F^-1*R*F*L*F*L^-1*F*L*F^-1*L*F*L*F^-1*R^-1*B^-1*L^-1*B*L*R*L^-2*R*F^-1*L^-1*F*R^-1*L^-1*F^-1*R^-1*F^-1*L^-\
1*F*R^-1*L^-1*R^-1*L*F^-1*(F^-1*L^-1)^2*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL\
*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*B^-1*L^-1*B*F^-1*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F\
^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*(R^-1*L)^2*F*L^-2*BR*F\
*BR^-1*L^-1*BL^-1*F*BL*F^-1*BR^-1*B*BR*L^-1*F^-2*L^-1*BR*R^-1*BR^-1*R*B*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*L*F*BR^-1*F^-1*R\
^-1*B^-1*L^-1*R^-1*BL^-1*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^-1*L^-1*F^-1*L*(L*F)^2*L*F^-1*(L*F)^2*L^-1*F^-1*L^-1*\
F*L*F*L^-1*F*L*F^-2*L^-1*F*L^2*F^-1*L^-1*F*L^-1*(F^-1*L)^2*(F*L^-1)^2*F^-1*L*F^-1*L^-2*(F^-1*L)^2*F*(L*F^-1)^2*(L^-1*F\
^-1)^2*L*F^-1*L^-1*F*L*F^-1*L*F*R^-1*L^-1*B^-1*L*B*R*L*F^-1*L^-1*F*L*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*F*(L^-1*F*L*F^-1)^\
2*L*F*L*F^-1*L^-1*F*L^-2*F*L*F^-1*L^-1*F^-1*L*F^-1*L^-1*(F*L)^3*F^-1*L^-1*F^-1*L*R^-1*B^-1*L^-1*B*L*R*L^-2*(R*F^-1*L^-\
1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*L*F*R*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*\
L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R\
^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*L*B^-1*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-\
1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^2*R*F^-1*L^-1*F*R^-1*L^-1*F^-1*L^-1*R^-\
1*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-2*L^-1*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*B\
R*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*(R^-1*L^-1)^2*F*L*R^2*F*R^-1*L*F^-1*R*B*R^-1*B\
^-1*(R^-1*F^-1)^2*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*L*BL*B*BL^-1*L*BR*R^-1*BR^-1*B^-2*R^-2*F^-1*\
BR*F*BR^-1*L*BL^-1*F*BL*F^-1*L*BR^-1*B*BR*L^-1*F^-1*R^-2*BR*R^-1*BR^-1*B^2*L*B^-1*L^-1*BR*R*BR^-1*R^-1*BL*BR*BL^-1*B*B\
R^-1*R*BR^-1*R^-1*B^-1*F^-1*BR^-1*L^-1)^2*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^-1*L^-1*F^-1*L*(L*F)^2*L*F^-1*(L*F)^\
2*L^-1*F^-1*L^-1*F*L*F*L^-1*F*L*F^-2*L^-1*F*L^2*F^-1*L^-1*F*L^-1*(F^-1*L)^2*(F*L^-1)^2*F^-1*L*F^-1*L^-1*F^-1*L*F*(L^-1\
*F^-1)^2*L*F^-1*L*F*(L*F^-1)^2*L^-1*F^-1*(F^-1*L^-1*F*L)^2*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*F^-1*(F^-1*L*F*L^-1)^2*\
F*L*F^-1*L^-1*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^2*L^-1*F^-1*L^2*F*L^-1*F^-1*L^-2*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1\
*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*L*F*R*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F\
^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^\
-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*L*B^-1*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*B\
R*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^3*R*F^-1*L^-1*F*R^-1*L^-1*B^-1*L*B*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L\
^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*\
F*R^-2*L^-1*F*L*R^2*F*R^-1*L*F^-1*R*B*R^-1*B^-1*(R^-1*F^-1)^2*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*\
L*BL*B*BL^-1*L*BR*R^-1*BR^-1*B^-2*R^-2*F^-1*R*BR*F*BR^-1*L^-1*F^-1*BL^-1*F^-1*R*BL*F^-1*BR^-1*B*BR*L^-1*F^-1*R^-1*BL*B\
^-1*BL^-1*R^-1*BR*B*L^-1*B^-1*F^-1*BR^-1*R^-1*B^-1*R*B*R^-1*BR*R*BR^-1*R^-1*BL*BR*B*BL^-1*L^-1*BR^-1*F*L*BL*B^-1*BL^-1\
*BR^-1*F^-1*L*F*L^-1*F^-1*(L*F^-1*L)^2*F*L^-1*F^-1*L^-1*R*F^-1*L^-1*F*R^-1*L^-1*F^-1*L^-1*R^-1*F^-1*L^-1*F*R^-1*L^-1*R\
^-1*L*F^-1*(F^-1*L^-1)^2*B^-1*L*B*R^-1*F*R^-1*(F^-1*R)^2*F*R^-1*L*R*B*R^-1*B^-1*R^-1*F^-1*R^-1*L*BR*F*BR^-1*L*BL^-1*F*\
BL*F^-1*L^-1*BR^-1*B^-1*BR*R*BR*R^-1*BR^-1*R*B*L^-1*B*BL*B^-1*BL^-1*F^-1*B^-1*BL^-1*L^-1*BL*L*BL*BR*BL^-1*B*BR^-1*R^-1\
*F^-1*BR^-2*L^-2*B^-1*F^-1*BR^-1*L^-1*F*(L*F^-1)^2*L^-1*F^-1*R^-1*F*L*F*L^-1*F^-1*R*F^-1*L^-1*F^-1*(L*F^-1*L^-1*F)^3*L\
*F^-1*L^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*F^-1*L*F*L^-1*F^-1*L*F^-1*L^-1*F^-1*L*F*L*(F^-1*L^-1)^2*F*(L^-1*F^-1)^2*L^-2*F*L*\
F*L^-1*R^-1*F*L*F*L^-1*F^-1*R*(L*F*L^-1*F)^2*(L*F^-1)^2*(L^-1*F)^2*L*F^-1*L*F*L^-2*F^-1*L*F*L^-1*F*L*F^-1*L^-1*F^-1*L^\
2*F*R^-1*L^-1*B^-1*L*B*R*L*F^-1*(L^-1*F)^2*F*L*F^-1*L^-1*(F^-1*L)^2*F*L^-1*R^-1*B^-1*L^-1*B*L*R*F^-1*L^-1*F^2*L^-1*F^-\
1*L*F*L^-1*F*L*F*L^-1*F^-1*L*F^-1*(L^-1*F^-1*L*F)^2*L^-1*F*L*F^-1*L^-1*F^-1*R^-1*F*L*F^-1*L^-1*F^-1*R*L*F^2*L^-1*F^-1*\
L^2*F*L^-1*F^-1*L^-2*(R*F^-1*L^-1*F*R^-2*F^-1*L^-1*F*R^-1*L^-1*R^-1*L*F^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*L*F*R\
*B*R^-1*B^-1*R^-1*F^-1*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*B\
R^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R^-1*F*BR*F*BR^-1*F^-1*BL^-1*F^-1*BL*R*BR^-1*B*BR*L^-1*F^-2*L^-1*BL*B*BL^-1*L*BL*L*B^-1\
*L^-1*R^-1*BL^-1*B*L^-1*B^-1*R^-1*BL^-1*L^-1*BL*F^-1*BL*BR*BL^-1*BR^-1*R^-1*F^-1*BL*F^-1*B^-1*BL^-1*F^-1*B^-1)^3*R*F^-\
1*L^-1*F*R^-1*L^-1*B^-1*L*B*R^-1*F*R^-1*F^-1*R*B^-1*L*B*L^-1*R^-1*F*R^-1*F^-1*R*F^-1*L^-1*R*F*R^-1*F^-1*BR*F*BR^-1*BL^\
-1*F*BL*F^-1*L*BR*F^-1*BR^-1*B*BR*R^-1*BR^-1*B^-1*F^-2*R*F*R^-2*L^-1*F*L*R^2*F*R^-1*L*F^-1*R*B*R^-1*B^-1*(R^-1*F^-1)^2\
*R*BR*F*BR^-1*L^-1*F^-1*BL*R^-1*BL^-2*F^-1*BL*BR^-1*B*BR*L*BL*B*BL^-1*L*BR*R^-1*BR^-1*B^-2*R^-2*F^-1*R*BR*F*BR^-1*L^-1\
*F^-1*BL^-1*F^-1*R*BL*F^-1*BR^-1*B*BR*L^-1*F^-1*R^-1*BL*B^-1*BL^-1*R^-1*BR*B*L^-1*B^-1*F^-1*BR^-1*R^-1*B^-1*R*B*R^-1*B\
R*R*BR^-1*R^-1*BL*BR*B*BL^-1*L^-1*BR^-1*F*L*BL*B^-1*BL^-1*BR^-1*L*F*L^-1*F^-1*R*F^-1*(L*F*R^-1)^2*F^-1*L^-1*F*R^-1*L^-\
1*R^-1*L*R*L*R^-2*F*R^-1*(F^-1*R)^2*F*R^-1*L*F*R*B*R*B^-1*R^-2*F^-1*R*BR*F*BR^-1*BL^-1*F*BL*F^-1*L*BR*F^-1*BR^-2*B*BR*\
L^-1*F^-2*L^-1*BL*B^-1*BL^-1*R^-1*BR*B*L^-1*B^-1*F^-1*BR^-1*B*L*B^-1*L^-1*BL^-1*L^-1*BL*L*BL*BR*B*BL^-1*L*F^-2*BL^-1*B\
^-1*BR^-1;
```




GAP doesn't know that you can just turn the puzzle upside down and even though it found a (relatively) short sequence for writing U moves in terms of the equatorial moves, it spits out a really long sequence for D moves.

*edit:* The move sequences produced above have to be read _in reverse order_. I dun goofed on defining which way pieces should cycle, because, you know, I'm not familiar with GAP and all.


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## DNF_Cuber (Jan 30, 2021)

Hi, I got my first FTO today, and I am having some difficulty with LBT(Last bottom triple)
I think I have the centers "Oriented" (W,O, R , and Grey centers are built, first two triples are solved, all edges are permuted) But I can't seem to figure out how to make the white-purple-green-red corner paired up with a purple and green triangle and put it in its spot.
EDIT: Now I somehow have the triple paired but upside-down in its slot
EDIT2: Finished LBT


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## PapaSmurf (Jan 30, 2021)

I'm glad you sorted it out. Keep on solving!


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## ProStar (Feb 21, 2021)

I recently got an FTO and I've solved it a couple times with the help of Ben's tutorial. However, I've found myself in a situation that I don't believe is covered in the tutorial. I have the entire puzzle solved, except for a single 2-swap of triangles, a green and yellow one specifically. Is this a possible state, and if so, how do I solve it? I've not taken it apart or messed with the puzzle at all, so the state should be possible.

There's a chance that a corner got twisted, in which case there would be a single triple left to solve. However, I don't know how to solve that state either.


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## OreKehStrah (Feb 21, 2021)

ProStar said:


> I recently got an FTO and I've solved it a couple times with the help of Ben's tutorial. However, I've found myself in a situation that I don't believe is covered in the tutorial. I have the entire puzzle solved, except for a single 2-swap of triangles, a green and yellow one specifically. Is this a possible state, and if so, how do I solve it? I've not taken it apart or messed with the puzzle at all, so the state should be possible.
> 
> There's a chance that a corner got twisted, in which case there would be a single triple left to solve. However, I don't know how to solve that state either.


I believe that’s fine. It’s basically going to need to swap this pieces and 2 LL triangles. There are algs you can use to solve those cases.


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## ProStar (Feb 21, 2021)

OreKehStrah said:


> I believe that’s fine. It’s basically going to need to swap this pieces and 2 LL triangles. There are algs you can use to solve those cases.



I can't seem to find an algorithm to solve a single flipped triple, and also can't figure it out intuitively. Do you have an alg you use for this case that you could provide?


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## Kit Clement (Feb 21, 2021)

A single triple flip with all else solved is an impossible state. Two triangles swapped is a possible state. Pair one of the triples and flip it. You'll likely have to break that triple up and form a different one to proceed, as you'll be in an odd TCP situation (one paired triple, one OPF triple).


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## ProStar (Feb 21, 2021)

Kit Clement said:


> A single triple flip with all else solved is an impossible state. Two triangles swapped is a possible state. Pair one of the triples and flip it. You'll likely have to break that triple up and form a different one to proceed, as you'll be in an odd TCP situation (one paired triple, one OPF triple).



Alright, I've solved it. Thanks!


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## ProStar (Feb 22, 2021)

What are the best algs for CP?


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## DNF_Cuber (Feb 22, 2021)

ProStar said:


> What are the best algs for CP?


I used the ones from the sheet in the description of Ben's vid. IDK about any others.


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## carcass (Feb 23, 2021)

I think Qiyi is probably working on the first good FTO. They made the first good clock, the first good squan, the best domino and 2x2x3. They tend to step in when hardware is garbage.


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## Kit Clement (Feb 23, 2021)

carcass said:


> I think Qiyi is probably working on the first good FTO. They made the first good clock, the first good squan, the best domino and 2x2x3. They tend to step in when hardware is garbage.



Unless QiYi's instagram lied a few months ago, your thought is wrong. Please don't make baseless speculation.



https://media.discordapp.net/attachments/737435950031831104/783163897913802752/image0.png?width=312&height=676


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## Kit Clement (Feb 23, 2021)

Hilariously enough, QiYi has announced what appears to be a Rex Cube, which is just an FTO shape mod:


__
http://instagr.am/p/CLoYh1VFBfl/


Doesn't seem so "imporssible" anymore


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## kits_ (Feb 25, 2021)

ProStar said:


> What are the best algs for CP?


I'm not sure if these are the best, but I use:
F' D F' U' F D' F' U F' (L face clockwise)
F U' F D F' U F D' F (L face counterclockwise)

(addition)
clockwise: 





Twizzle Editor ᴬᴸᴾᴴᴬ







experiments.cubing.net




counterclockwise: 





Twizzle Editor ᴬᴸᴾᴴᴬ







experiments.cubing.net


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## Kit Clement (Feb 25, 2021)

kits_ said:


> I'm not sure if these are the best, but I use:
> F' D F' U' F D' F' U F' (L face clockwise)
> F U' F D F' U F D' F (L face counterclockwise)



Just wanted to mention that this is for the less common notation where U/F share an edge rather than a corner.


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## rubik2005 (Mar 28, 2021)

So over the span of a couple of days, I tried solving the FTO without any tutorials or reference, and I have made it to the last layer specifically PLL (if that's a step). Basically three center (1-color pieces) need to rotate counter-clockwise, and three edges (2-color pieces) need to rotate clockwise, so I wanted to know if there's an alg that can solve this case:


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## TipsterTrickster (Mar 28, 2021)

rubik2005 said:


> So over the span of a couple of days, I tried solving the FTO without any tutorials or reference, and I have made it to the last layer specifically PLL (if that's a step). Basically three center (1-color pieces) need to rotate counter-clockwise, and three edges (2-color pieces) need to rotate clockwise, so I wanted to know if there's an alg that can solve this case:


there aren't any current speedsolving methods that finish with this kind of 'pll', so we don't really have good algs for it, but you can solve it with commutators, and an edge alg.
edge alg: hold the corner in front btw,
F U F' U F U F' U (cycles U layer edges cw)
as for the rest you can use center comms like, again corner in front and U on top.
R' Rw B L' B' Lw L' B L B'


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## thelargeman2048 (Apr 1, 2021)

ive got an fto and i can get one side but thats about it. if anyone knows any beginner friendly fto tutorials thatd be very helpful


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## Kit Clement (Apr 1, 2021)

thelargeman2048 said:


> ive got an fto and i can get one side but thats about it. if anyone knows any beginner friendly fto tutorials thatd be very helpful



See the video in the first post of this thread.


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## lucazdoescubingstuff (Jan 5, 2023)

sorry for replying to an old thread but, should i get an FTO? seems fun, and I could compete in a new event in the weekly comps, so should I?


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## PapaSmurf (Jan 6, 2023)

Get it, it is! And I should probably compete in the weekly comp too.


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