A Collection of Cubing Curiosities

Discussion in 'General Speedcubing Discussion' started by macky, Aug 23, 2011.

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

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

2. KirjavaColourful

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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']

Last edited: Aug 23, 2011
3. qqwrefMember

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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.

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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.

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.

How do you use this?

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

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

Last edited: Aug 25, 2011
5. sa11297Member

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

6. Lucas GarronSuper-Duper ModeratorStaff Member

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.

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

Last edited: Aug 23, 2011
7. StefanMember

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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.

8. StefanMember

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9. irontwigMember

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

10. CubenoviceForever Slow

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

11. clementMember

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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...

12. Robert-YSuper ModeratorStaff Member

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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

13. ForteMember

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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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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.

Last edited: Aug 23, 2011
15. qqwrefMember

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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 )

16. irontwigMember

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I hope to cancel my membership soon.

17. KirjavaColourful

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

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.

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

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,

That is kind of neat.

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?

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.

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

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?

Last edited: Aug 24, 2011
19. Lucas GarronSuper-Duper ModeratorStaff Member

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).

(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).

Last edited: Aug 24, 2011

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