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A Collection of Cubing Curiosities

macky

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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
* Anything else relating more or less to the cube itself that is particularly interesting.

Please send suggestions to the author, with attribution (name and date) when known.

Thanks to Clément Gallet and Stefan Pochmann for the inspiration.
Please suggest inclusions. There must be some real gems in A Collection of Algorithms.
 

Kirjava

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#2
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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#3
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'UFU2RUR'U.
 

macky

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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?


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'UFU2RUR'U.
Fixed, thanks. I'll look at the others when I have a cube.
 
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Lucas Garron

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#6
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.

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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#11
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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#12
Originally from this thread: http://www.speedsolving.com/forum/showthread.php?26442-Hardest-Scramble-ever

What's God's number for LL? Anyone know it?
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 :p
 
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#15
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 :p)
 
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#16
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 :p)
I hope to cancel my membership soon.
 

Kirjava

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#17
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'

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.

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.
 

macky

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Added some. Pointing out non-examples.

Unexpected U perm found by me: (M'U2M) U (M'U2M) U (M'U2M)
Unexpected, but not surprising since the algorithm clearly leaves EPLL.

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.

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 :p
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:
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.

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.

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.
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

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#19
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. :p

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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#20
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"...

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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