# "Shortest" PLL Algs?



## qqwref (Oct 14, 2013)

What are the shortest possible PLL algs, in terms of how long they are when written down? This should be more interesting than just finding move-optimal stuff, and I think there's a lot of room for creativity. The most common algs may not be the shortest!

To start with, the following notations are OK:
- [P,Q] is a commutator (that is, P Q P' Q')
- [P:Q] is a setup move/conjugate (that is, P Q P')
- (P)n is P repeated n times; parentheses are unnecessary if it's clear what P is
- Any face or slice moves, rotations, or lowercase moves.
Also, the algorithm can do the PLL on any face, and you do not have to include any adjustment at the end, or rotations at the very start/end.


So here's a set of pretty short algs. Think you can do any better? 
A: R2[RFR',B2]R2, R2[B2,RFR']R2
E: (R'UL'D2LU'RU2)2
F: R'URU'R2F'U'FU[R,F]R2
G: [RL:U2]yL'UR'U2LU'R, R'UL'U2RU'Ly[RL:U2], L'R'U2LRyLU'RU2L'UR', LU'RU2L'UR'yR'L'U2RL
H: (M2U)6
J: R'L'[d2,RUR']RL, [RL:[d2,R'U'R]]
N: (r'DrU2)5, (rDr'U2)5
R: R[U2R'U2,UR'F'R]R', R'[U2RU2,U'RBR']R
T: [R2D':F2][B2D:L2]
U: M2uMu2MuM2, B2UMU2M'UB2
V: [F'UBU'F:U][U2,B]
Y: F2[DR2:U][R'U'R:F2]
Z: M2uM2D'MS2M'


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## Stefan (Oct 14, 2013)

How do you measure the length?


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## qqwref (Oct 14, 2013)

Number of characters, or equivalently string length. The length of "(M2U)6" is 6.


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## Stefan (Oct 14, 2013)

So my efforts of finding out that "L" is the shortest capital letter in this forum's font were in vain...

A: L'[F,R'B2R]L (12)
J: L'[R'd2R,U]L (12)


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## Ranzha (Oct 14, 2013)

Ga: [R2F2:[B2,U][B2,D]]
Just inverse and mirrors for the other HTM-optimal Gs.


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## qqwref (Oct 14, 2013)

Well this is embarrassingly obvious, but...
[RL:U2]yL'UR'U2LU'R => [RL:U2]F'UB'U2FU'B
R'UL'U2RU'Ly[RL:U2] => R'UL'U2RU'L[BF:U2]
L'R'U2LRyLU'RU2L'UR' => L'R'U2LRFU'BU2F'UB'
LU'RU2L'UR'yR'L'U2RL => LU'RU2L'UR'B'F'U2BF


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## Christopher Mowla (Oct 14, 2013)

This is a little off topic: parity algs!

*Adjacent Double Parity*
[r2B'RwB:r(rU2)4] (17)(WCA)

*3 flip which messes one F3L slot and LL (not double parity)*
rU2[BU2B': (U2r)3]r (18)(WCA)

*Opposite Double Parity*
[r2UF2U':r][UF2U,r] (19)(WCA)

*One edge flip (not double parity)*
[F2r2:[F2,U2l]]r[U2,r'] (23)(WCA or SiGN)(Frédérick Badie modified)
[l2B2:[B'l'B,U2][rB2:l]] (24)(WCA or SiGN)(cmowlaparity)
[l2B2:[Bl'B',U2][rB2:l]] (24)(WCA)(cmowlaparity's cousin)
[r2B2:[U2,lU2r'][lF2:r]] (24)(WCA or SiGN)("Standard Alg")
[l2B2:[U2,rU2l'][rB2:l]] (24)(WCA or SiGN)
[led:[d,b'2R'b][2Rd:2R']] (25)(SiGN)
[U2,lU2r'][F2r:F2]r(r2F2)3 (26)(WCA or SiGN)


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## cannon4747 (Oct 14, 2013)

I found this a while ago using commutators and conjugates but idk how to write it down in any other way.

E = R'D'RLDL'R'D'RU2R'D'RLDL'R'D'R


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## TDM (Oct 14, 2013)

cannon4747 said:


> I found this a while ago using commutators and conjugates but idk how to write it down in any other way.
> 
> E = R'D'RLDL'R'D'RU2R'D'RLDL'R'D'R


Yours is (R'D'RLDL'R'D'RU2)2
Not exactly the same, but similar:
[R'D'RLDL'R'D'R,U2]

EDIT:
[noparse][[RU'L2],U2][/noparse]
Shortest E perm on here.

EDIT2:
Similar thing for a short A perm:
[noparse][[RU'L2],U][/noparse]


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## Stefan (Oct 14, 2013)

TDM said:


> [[RU'L: D2],U2]
> Shortest E perm on here. I have to have a space in so it doesn't show as .



[noparse][[RU'L2],U2][/noparse]


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## TDM (Oct 14, 2013)

Stefan said:


> [noparse][[RU'L2],U2][/noparse]


I didn't know you could use those to stop smileys.


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## whauk (Oct 14, 2013)

Z-perm: (UF2)6M'U2M
1 charcter shorter than qq


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## Renslay (Oct 14, 2013)

Supercube centers:

Rotate the U center with 180 degrees (like U2):
[[RL:U2]:U]U

Rotate the U center and L center with 90 degrees (like U and L'):
[U,MEM']


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## TDM (Oct 14, 2013)

Renslay said:


> Rotate the U center with 180 degrees (like U2):
> [[RL:U2]:U]U


(L'ULU)5
(LUL'U)5
(RUR'U)5
(R'URU)5


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## Renslay (Oct 14, 2013)

TDM said:


> (L'ULU)5
> (LUL'U)5
> (RUR'U)5
> (R'URU)5



Damn. 

Playing around, I found an interesting U perm (not the shortest):
(R2d)12[D2,S]


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## TDM (Oct 14, 2013)

Renslay said:


> Damn.
> 
> Playing around, I found an interesting U perm (not the shortest):
> (R2d)12[D2,S]


I first learned the third one when someone at a comp showed it to me, but I do the first one sometimes if I have a cube in one hand. The other 2 are basically the same thing.


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## qqwref (Oct 15, 2013)

cmowla said:


> This is a little off topic: parity algs!


Haha, that's fine. Very cool stuff!



TDM said:


> [noparse][[RU'L2],U2][/noparse]
> [noparse][[RU'L2],U][/noparse]


Nice, that does save some characters on the E perm (and also: [[RU'L:d2],U] and [[R'UL':d2],U] for J). The A perm is only as good as the one I have though.

Another short A perm: R[B,[F,R']]R' and inverse.

And some G perms based on the same idea: [RL:U2][F'UB':d2], [R'UL',d2][BF:U2], L'R'U2LR[FU'B,d2], [LU'R,d2]B'F'U2BF



whauk said:


> (UF2)6M'U2M


Very cool, definitely would not have thought of that.


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## TDM (Oct 16, 2013)

What about doing this but with fewest letters instead of fewest characters? So A perms ([R2:[B2,[R:F]]] and [R2:[[R:F],B2]]) would count as 4 etc.


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## qqwref (Oct 18, 2013)

TDM said:


> What about doing this but with fewest letters instead of fewest characters? So A perms ([R2:[B2,[R:F]]] and [R2:[[R:F],B2]]) would count as 4 etc.


Thought I'd give this a shot. Obviously in this case inverses and mirrors are the same number of letters, so I don't bother listing them separately.

[noparse]A: [L:[[R2],U']] = 4
E: [[RU'L2],U2] = 5
F: [R', U][R2:[F',U'][R,F]] = 7
G: [R2F2:[B2,U][B2,D]] = 6
H: (M2U)6 = 2
J: [L':[[R':d2],U]] = 4
N: (r'DrU2)5 = 4
R: [R:[[U2:R'],U[R':F']]] = 6
T: [R2D':F2][B2D:L2] = 6
U: [R2U':[S',U2]] = 4
V: [[F'U:B]:U][U2,B] = 6
Y: F2[DR2:U][[R':U']:F2] = 7
Z: [M2,U](E2M')2 = 4[/noparse]


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## Christopher Mowla (Oct 20, 2013)

With TDM's suggested metric,

Here's an algorithm which is one move less than the other algorithms I have already listed for qqwref's metric for the visually pure single dedge flip case:
[(rU2)2:r(r[U(RwU2)2:U2])2] = 8, where the Rw move includes the central slice on odd cubes,

...just in case anyone was interested.


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