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FTO thread

OreKehStrah

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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!
 

PapaSmurf

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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.
 

Kit Clement

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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.
 

PapaSmurf

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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.
 

qwr

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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.
 
Last edited:

zslane

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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.
 

qwr

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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.
 

xyzzy

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

Code:
				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.)

Code:
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.

Code:
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:
Code:
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.
 
Last edited:

DNF_Cuber

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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
 
Last edited:

ProStar

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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.
 

OreKehStrah

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May 24, 2019
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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.
 

Kit Clement

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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).
 

ProStar

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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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