# A Collection of Cubing Curiosities



## macky (Aug 23, 2011)

http://cubefreak.net/other/curiosities.php



> The standard T-perm RUR'U'R'FR2U'R'U'RUR'F' with R/R' replaced by r/r' is an A-perm. This is (probably) useless for speedsolving, but it's interesting nonetheless. The purpose of this page is to collect and preserve these cubing curiosities, which until now have existed as "folklore" with no proper home.
> 
> Considered for inclusion are
> * Interesting algorithms, fingertricks, or ideas on the 3x3 orother twisty puzzles, not necessarily of any practical use, and especially oddities applying to one or very few cases and with no obvious explanation
> ...



Please suggest inclusions. There must be some real gems in A Collection of Algorithms.


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## Kirjava (Aug 23, 2011)

rUR'U'r'FR2U'R'U'RUR'F' - T Perm derived OLLCP

EDIT: Not too sure of the criteria for things to go into this list. I mean, there's a plethora of magic LSE stuff that most would be uninterested in (if they even understand it).

EDIT2: Maybe this is notable; X Y X' Z Y' Z' = [Z: [Z' X, Y]] = [X:Y] [Z:Y']


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## qqwref (Aug 23, 2011)

A superflip (independently invented by me and many others): ((M'U)4 x y')3

<R,U> 2-gen M'U2MU2 (also an intuitive 2gen 3-cycle) found by me: (R U R2 U' R') (U' R' U2 R U)

Unexpected U perm found by me: (M'U2M) U (M'U2M) U (M'U2M)

And LUL'x'U'F'U'FU2RUR'U should be LUL'x'U'F'*U*FU2RUR'U.


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## macky (Aug 23, 2011)

Kirjava said:


> rUR'U'r'FR2U'R'U'RUR'F' - T Perm derived OLLCP


I wouldn't count this since it's not surprising that a (partially) thick version of a last layer algorithm gives another last layer algorithm. When the result is another PLL, it's a bit more magical.



> EDIT: Not too sure of the criteria for things to go into this list. I mean, there's a plethora of magic LSE stuff that most would be uninterested in (if they even understand it).


Could you list some examples? Yeah, I couldn't think of good non-examples, so I'll try to write down more precise criteria after some posts.



> EDIT2: Maybe this is notable; X Y X' Z Y' Z' = [Z: [Z' X, Y]] = [X:Y] [Z:Y']


How do you use this?




qqwref said:


> A superflip (independently invented by me and many others): ((M'U)4 x y')3


Yeah, this one is simple and well known but surprising and beautiful enough to deserve a spot.



> And LUL'x'U'F'U'FU2RUR'U should be LUL'x'U'F'*U*FU2RUR'U.


Fixed, thanks. I'll look at the others when I have a cube.


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## sa11297 (Aug 23, 2011)

R2DR'U2RD'R'U2 R2FRB'R'F'B
an e perm that I made up


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## Lucas Garron (Aug 23, 2011)

You probably meant (LU'Ru2L'UR')2 for the E-perm. Also, most of your algs should work straight in alg.garron.us if you'd like to link.



sa11297 said:


> R2DR'U2RD'R'U2 R2FRB'R'F'B
> an e perm that I made up


 
You're missing a move. Also, that does not seem curious to me at all. It's just a concatenation of two OLLs which flow together by one move.


Typo: "orother"

Also popularized by Chris: The number of unsolved states of a 3x3x3 is prime.

Another "beautiful" alg: (R'U'RU')5
Also (R U R' F)5


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## Stefan (Aug 23, 2011)

Nice page/thread...

Correction: My cube-in-cube rotates the cube around the ULF corner, not the URF corner.

(R U R' U') (L' U' L U) (U R U' R') (U' L' U L) is a 3-cycle of *edges *on the 3x3x3 and a 3-cycle of *corners *on the megaminx.


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## Stefan (Aug 23, 2011)

(R U R' U' y)7 is a nice pattern and can also be created by getting four center-edge pairs into one slice and turning it ([R B L R' F R : E]) or getting four corner-edge pairs into one layer and turning it ([R B L R' F R : D] U'). Note that the setup R B L R' F R is the same for both (never realized that until right now...).

((R U R' d')7 and R B M' U R are "shorter" but don't look as nice)


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## irontwig (Aug 23, 2011)

All 10 move 2C2E swaps are cyclic shifts and/or inverses of each other.


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## Cubenovice (Aug 23, 2011)

In FMC the best results are typically optained by insertion of some moves.
Both (tied) world records did not use insertions...


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## clement (Aug 23, 2011)

Stefan said:


> Correction: My cube-in-cube rotates the cube around the ULF corner, not the URF corner.


Another way to do the cube-in-cube : (R F2 L R F2 L' R2 [u'] [r'])4 (found by Per Kristen Fredlund). [u'] [r'] is turn around URF corner.

Also : (R U' R' U [r] )3 flips 3 corners and 2 edges, so (R U' R' U [r] )6 flips 3 corners and (R U' R' U [r] )9 flips 2 edges.

On the contrary, (R U' R' U [u'] [r'])3 = id

In HTM, shortest non trivial id alg (one exemple) : R2 F R F' U' R2 B' R' B U
Correction : well, U2 D2 R2 L2 U2 D2 R2 L2 is shorter...


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## Robert-Y (Aug 23, 2011)

Originally from this thread: http://www.speedsolving.com/forum/showthread.php?26442-Hardest-Scramble-ever



Toad said:


> What's God's number for LL? Anyone know it?





Rinfiyks said:


> It's 16 I believe.
> 
> Edit: F' L2 B L B' U2 B L' B' L2 U F U' R U2 R'
> Edit2: Thanks to this page. Just had to write a little program to find the longest algorithm. It's the only 16 move algorithm, interestingly.



Here's another one:

The fastest known diagonal corner swap on a 333, which preserves CO, is this E perm: http://www.youtube.com/watch?v=qCG6bNLqUkE

Even if we could ignore all edges and centres, there's nothing better afaik


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## Forte (Aug 23, 2011)

It's kinda arbitrary, but the Sune is a well known algorithm so I thought I might as well post this.

R U R' U R U2 R' = [R U R2 : R U2 R2]


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## cmhardw (Aug 23, 2011)

Lucas Garron said:


> Also popularized by Chris: The number of unsolved states of a 3x3x3 is prime.


 
Ooooh, Lucas that reminds me! The number of unsolved states of the 8x8x8 and 11x11x11 are also prime numbers (3+8=11 is how I remember that). I'll send that off to Macky, if someone hasn't sent it already.


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## qqwref (Aug 23, 2011)

This is definitely a curiosity - the "31 club" in FMC... A surprisingly large number of famous/important cubers have 31 moves as their official personal best. The current list includes Dan Cohen, Gilles Roux *and* Lars Petrus, Lucas Garron and me, Mike Hughey, Stefan Pochmann, and Yu Nakajima. (And it was the NAR twice )


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## irontwig (Aug 23, 2011)

qqwref said:


> This is definitely a curiosity - the "31 club" in FMC... A surprisingly large number of famous/important cubers have 31 moves as their official personal best. The current list includes Dan Cohen, Gilles Roux *and* Lars Petrus, Lucas Garron and me, Mike Hughey, Stefan Pochmann, and Yu Nakajima. (And it was the NAR twice )


 
I hope to cancel my membership soon.


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## Kirjava (Aug 23, 2011)

clement said:


> In HTM, shortest non trivial id alg (one exemple) : R2 F R F' U' R2 B' R' B U



Here's one that uses every face turn; D F R U2 B L U' L' U2 B' R' U F' D'



macky said:


> Could you list some examples? Yeah, I couldn't think of good non-examples, so I'll try to write down more precise criteria after some posts.



I think the most delicious thing regarding LSE tricks is how you can give yourself easy orientations by changing the definition of orientation and misorienting centres. 

For example, on this scramble; MURUR'U'M2URU'r'

orientation would be solved with this (or similar); M'U2M'U2M'UM'

with the alternative definition, orientation for this case is simply; M'

However, an explanation of this technique would not be very concise.



macky said:


> How do you use this?



X Y X' Z Y' Z' = X Y X' *Y' Y* Z Y' Z'

[M', RUR'U'] = FU -> RU -> BU
[RUR'U', E'] = BU -> BR -> RF

X - M', Y - RUR'U', Z - E'

M' RUR'U' M E' URU'R' E = FU -> RU -> BU -> BR -> RF

It's really only useful for K4 LL so far. Doing intuitive 5-cycles in BLD with it is... quite hard.

Some other stuff..

This is pretty neat; [FRBL,U]

The shortest alg that affects orientation with no effect on permutation is; RUR2FRF2UFU2

RU'r'U'M'UrUr' is a bit magical for LSE - much shorter/faster than any <MU> one (somewhat unique in this respect), but doubt this warrants inclusion in the list.


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## macky (Aug 24, 2011)

Added some. Pointing out non-examples.



qqwref said:


> Unexpected U perm found by me: (M'U2M) U (M'U2M) U (M'U2M)


Unexpected, but not surprising since the algorithm clearly leaves EPLL.



clement said:


> In HTM, shortest non trivial id alg (one exemple) : R2 F R F' U' R2 B' R' B U
> Correction : well, U2 D2 R2 L2 U2 D2 R2 L2 is shorter...





Kirjava said:


> The shortest alg that affects orientation with no effect on permutation is; RUR2FRF2UFU2


These seem borderline to me. They're certainly facts to be noted, but of course there is some shortest alg in each case. It'd be surprising if that algorithm is particularly cute. Like,



Kirjava said:


> This is pretty neat; [FRBL,U]


That _is_ kind of neat.



Robert-Y said:


> The fastest known diagonal corner swap on a 333, which preserves CO, is this E perm: http://www.youtube.com/watch?v=qCG6bNLqUkE
> 
> Even if we could ignore all edges and centres, there's nothing better afaik


Again seems borderline for the same reason. I'll wait on inclusion for now.

But more importantly, ", which preserves CO," should read "that preserves CO" (without commas). With that kind of grammar, how's a non-cuber to know that not every diagonal corner swap on a 3x3 preserves CO?



cmowla said:


> [snip]


I don't see how anything you wrote is particularly surprising or beautiful. The derivations are impressive but don't seem miraculous in any way. The formulas are some ugly mess that follow by basic combinatorial considerations.



Forte said:


> It's kinda arbitrary, but the Sune is a well known algorithm so I thought I might as well post this.
> 
> R U R' U R U2 R' = [R U R2 : R U2 R2]



I don't see how that's surprising or illuminating. The conjugation looks arbitrary.



Robert-Y said:


> Rinfiyks said:
> 
> 
> > It's 16 I believe.
> ...



Could you clarify this? Of course God's number for LL is some number. Is it that there's a single case (up to some appropriate equivalence) with this distance? That might not be that surprising. Is it that, for this single case, there's a single algorithm (again up to equivalence) of length 16?


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## Lucas Garron (Aug 24, 2011)

macky said:


> I don't see how that's surprising or illuminating. The conjugation looks arbitrary.



I think he meant to write [R U R2, R U2 R2]. The Sune is one of the most important algs in cubing, and it is curious that the alg itself can be written as a commutator (because it's not obvious at a glance).
See http://www.speedsolving.com/forum/showthread.php?5783-Math-Problem-12

(Contrast with an even alg like UR, which can not be written as a commutator – but which is the same permutation as some algs which are commutators)


Correction: "The current list includes Dan Cohen, Gilles Roux *and* Lars Petrus, Lucas Garron, *Michael Gottlieb*, Mike Hughey, Stefan Pochmann, and Yu Nakajima."


Also, I'm not so sure about some of your attributions. But maybe I'm just jealous I didn't suggest them first. 

Regarding Hardwick's conjecture: I checked up to n=54 over four years ago when Chris brought up the n=3 constant; see this post; I don't recall Michael posting on this, but maybe I'm just missing something.

I've been showing people the RU-gen S'U2SU2 for years, although I just consider it common knowledge.
One other interesting variant (I use it all the time during BLD) is R2 U R U *R2'* U' R' U' R' *U2* R'.

In a similar vein, an Mu-gen U-perm where all the M-moves and all the u-moves can naturally go in the same direction each time: M2 u' M' u2' M' u' M2'


From an email to you (Macky) on June 4:

There are 5 (RU-gen) F2L cases where the corner is facing up. Four of these have very nice algs, while the fifth somehow does not. Moreover, each of the four algs is its own self-inverse (by permutation), *and* preserves the orientation of all other LL pieces. I think that's pretty curious; I can partially of explain it from the algs, but I don't quite see why it should work out this way.

RU'R'URU2'R'U'RUR' (edge in slot)
URURU2'R'U'RU'R2' (edge clockwise once from corner)
RU'R'U2RUR' (edge clockwise twice from corner)
[No good alg] (edge clockwise three times from corner)
R U2 R' U' R U2 R' U R U2 R'  (edge clockwise 4 times from corner)


Maybe not so exciting, but: There are exactly two states in the center of the cube group, and they are the closest and farthest possible from solved (in HTM).

The double-Sune can be rotated/mirrored/inverted to perform any 3-cycle of edges for a given corner orientation (keeping corners permuted).


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## Robert-Y (Aug 24, 2011)

macky said:


> Again seems borderline for the same reason. I'll wait on inclusion for now.
> 
> But more importantly, ", which preserves CO," should read "that preserves CO" (without commas). With that kind of grammar, how's a non-cuber to know that not every diagonal corner swap on a 3x3 preserves CO?



Sorry, I couldn't really find a nice and easy to understand way of describing this "oddity", thanks. I'm never sure when to use "which" or "that"...



macky said:


> Could you clarify this? Of course God's number for LL is some number. Is it that there's a single case (up to some appropriate equivalence) with this distance? That might not be that surprising. Is it that, for this single case, there's a single algorithm (again up to equivalence) of length 16?


 
Ah sorry. I thought Rinfiyks meant this is the only LL case that requires 16 moves to solve and all of the other LL cases require a maximum of 15 moves. I don't actually know myself.


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## Robert-Y (Aug 24, 2011)

I noticed this one on your webpage:

"An E-perm: (LU'Ru2L'UR')2. This is N-perm with u2 instead of U2."

If you replace U2 with E2, it becomes a Z perm.


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## TMOY (Aug 24, 2011)

And LU'Ru2L'UR' LU'RU2L'U(LE2L')R' is a V-perm.


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## cmhardw (Aug 24, 2011)

Lucas Garron said:


> Also, I'm not so sure about some of your attributions. But maybe I'm just jealous I didn't suggest them first.
> 
> Regarding Hardwick's conjecture: I checked up to n=54 over four years ago when Chris brought up the n=3 constant; see this post; I don't recall Michael posting on this, but maybe I'm just missing something.


 
Lucas, my apologies. I likely misremembered who had done the analysis (I'm the one who mentioned Michael Gottlieb in my post to Macky). Michael, did you do any analysis on this as well? Apologies if I have not given credit where credit is due, that was not my intention.


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## Kirjava (Aug 24, 2011)

LL Scramble; B2 D' F R2 F' D B U2 B U y2

OLL; R U R' U'

Feliks found this one.


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## cmhardw (Aug 24, 2011)

Kirjava said:


> LL Scramble; B2 D' F R2 F' D B U2 B U y2
> 
> OLL; R U R' U'
> 
> Feliks found this one.


 
I know it's been said already, but Feliks' technique is amazing!


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## DavidWoner (Aug 24, 2011)

Kirjava said:


> LL Scramble; B2 D' F R2 F' D B U2 B U y2
> 
> OLL; R U R' U'
> 
> Feliks found this one.


 
Erm, what do you mean he found it? This is an old trick for FM. It also applies to L R F2 R B L' U2 L B' R2 F L' which is oriented with r U' L' as well others I'm sure.


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## macky (Aug 25, 2011)

Added more. Thanks for your contributions.



Lucas Garron said:


> I think he meant to write [R U R2, R U2 R2]. The Sune is one of the most important algs in cubing, and it is curious that the alg itself can be written as a commutator (because it's not obvious at a glance).
> See http://www.speedsolving.com/forum/showthread.php?5783-Math-Problem-12


ah, I see. Did someone show that the commutators themselves are closed under composition? I remember reading something like this. In that case, I would mention the Sune is a representative non-obvious example.



DavidWoner said:


> Erm, what do you mean he found it? This is an old trick for FM. It also applies to L R F2 R B L' U2 L B' R2 F L' which is oriented with r U' L' as well others I'm sure.



I'd be interested in seeing some more.




Lucas Garron said:


> Also, I'm not so sure about some of your attributions. But maybe I'm just jealous I didn't suggest them first.


Yeah, for things that multiple people have known for a while, I want to replace the "suggested by" with just a nameless link to the source post. On the other hand (not really a reply to your comment), I think "popularized by" should stay; if Feliks really did popularize that one case enough case to turn that trick into common knowledge, then he deserves a mention, like with Rowe's E perm. In any case, speak up if there's any doubt.


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## qqwref (Aug 25, 2011)

cmhardw said:


> Lucas, my apologies. I likely misremembered who had done the analysis (I'm the one who mentioned Michael Gottlieb in my post to Macky). Michael, did you do any analysis on this as well? Apologies if I have not given credit where credit is due, that was not my intention.


I don't remember doing any analysis on this.


Another curiosity: certain algs can be done on only the top face of a bigger cube simply by replacing some of the moves. For instance, RwURw'U'Rw'FRw2U'Rw'U'RwURw'F'. The canonical Y perm works too, and Fw2UM'U2MUFw2. This is somewhat surprising since many common/fast algs do not do this. The R'U2RU2 R perm doesn't work, or the 2-gen U perm and the RU2R' style J perm doesn't although the RUR'F' one does. The MU H-perm works, the 2-gen one *almost* works (it does the intended effect plus two edge 2-cycles), and the RLU2 one almost works as well if you do it right. I'm still not really sure why some algs work and others don't, but it's definitely curious.

EDIT: Oh yeah, something else that I think I discovered: any PLL (any ZBLL, actually) can be done completely 2-gen plus exactly one F and one F'. The same thing occurs for D/D', B/B', and L/L'.


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## Athefre (Aug 25, 2011)

Kirjava said:


> RU'r'U'M'UrUr' is a bit magical for LSE - much shorter/faster than any <MU> one (somewhat unique in this respect), but doubt this warrants inclusion in the list.


 
Are you including recognition time when you say "faster"? Because surely, when using freecenters, the five move MU'M'UM' (UL/UR at UL and DF) and M'U'M'UM' (UL/UR at UL and UB) can be performed faster than the nine move RrUM. Even M'U2M'UMUM'UM' (UL/UR at UF and UB), which is the longest freecenter six-flip, is pretty fast.

Including recognition time though, it may be quicker to use your RU'r'U'M'UrUr' instead of searching for UL and UR to save four moves.

I hate to not add anything to the topic. I tried, but nothing came to mind.


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## Kirjava (Aug 25, 2011)

Yeah, I wouldn't rely on that. I only misorient centres when I happen to have recog'd white/yellow during CMLL and know a trick for the case.


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## Lucas Garron (Oct 10, 2011)

Hmm, I don't think this ever came up here: G = <UF, RBL> (Rokicki's proof)

From the same thread: A set of two random cube elements is about 50% likely to generate the entire cube group.


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