PBL

2019-10-19T18:50:55Z

AkiraSan: Fixed typo in an algorithm

'''Permutation of both layers''', abbreviated as '''PBL''', is a [[step]] used in a number of [[2x2x2]] methods (and thus [[corners first]] methods). Specifically, the [[Ortega method]] and the [[Guimond method]], perhaps the two most popular advanced [[2x2x2]] methods, both finish with a PBL step. When solving only corners, PBL has five unsolved cases, which can be recognized by the number of pairs on each side that are correctly permuted (a solved side has 4 pairs, an adjacent swap has 1 pair, and an opposite swap has 0 pairs), so the unsolved cases are 0+0, 1+0, 1+1, 4+0, and 4+1. [[Algorithm]]s for PBL (for 2x2x2 or CF) can be found at the [[Template:CxLLnav|CxLL]] pages (the grey cases in the topmost line).

It is actually possible, in theory, to do PBL in one step for [[3x3x3]] or [[Square-1]] - in this case we would be permuting not only corners but the whole layers. No [[speedcubing]] method actually uses this, though, because there are about 800 cases in total and recognition is very difficult. For the [[Square-1]], many methods do PBL in two steps, by first solving the [[corner]]s of both layers and then doing the [[edges]] of both layers. Even though two layers are being permuted at once, though, solvers conventionally refer to the steps as simply [[CP]] and [[EP]].

== Algorithms ==
<table cellspacing='5' cellpadding='5'><tr>
<td valign='top'>
==== Swap UF Corners ====

[[Image:PBL1.gif]]
* y' x U2 R' U' R U2 L' U R' U' R2
* (R' F R') B2 R F' (R' B2 R2) ([[PLL#A_Permutation_:_a|A Permutation]])
* U' x' R2 U2 R' U' R U2 L' U R'
* y' R2 F2 R' U' R F2 R' U R'
* y' (U') R' U L' U2 R U' R' U2 R L
* y' R U R' U' R' F R2 U' R' U' R U R' F' ([[PLL#T_Permutation_:_t|T Permutation]])
* y' R U2 R' U' R U2 L' U R' U' L
* y2 R' U L' U2 R U' R' U2 R2 B' (Ja Permutation)
* y' R2 U R2' U' R2 y' R2' U' R2 U R2 U'

</td>

<td valign='top'>
==== Swap U-Layer Diagonal ====
[[Image:PBL2.gif]]
* (R U' R' U') F2 (U' R U R') D R2
* (R U' R' U') F2 (U' R U R' U) F2
* L' U L D R2 D R' U' R D' R2
* F R U' R' U' R U R' F' R U R' U' R' F R F' ([[PLL#Y_Permutation|Y Permutation]])
* R' U L' U2 R U' x' U L' U2 R U' L
* [R' U R' U'] y [R' F'] [R2 U' R' U] [R' F R F]
</td>

<td valign='top'>
==== Swap UF + DF Corners ====
[[Image:PBL3.gif]]
* R2 U' R2' (U2' + y) R2 U' R2'
* R2' U' R2 U2' F2 U' R2
* R2' U' R2 U2' x' U2 x' U' R2
* R2 U F2 U2 R2 U R2
* y2 R2 U' B2 U2' R2' U' R2
* z' U2 L U2 (L' R') U2 R U2
</td>

</tr>
<tr>

<td valign='top'>
==== Swap U + D Diagonal ====
[[Image:PBL4.gif]]

* R2' F2 R2
* R2 B2 R2'
* (R L) U2 (R' L')
</td>

<td valign='top'>
==== Swap UB Corners + D-Layer Diagonal ====
[[Image:PBL5.gif]]

* [R U' R] F2 [R' U R']
* [R U' L] U2 [R' U R']
* y2 [R' U L'] U2 [R U' L]
* x2 y' R2 U' R2 U R2 U' R2 U R2
* x2 R D' L U2 L' D R'
* x2 R D' L U2 R' F U'
</td>

<td valign='top'>
==== Swap UB + DB Corners ====
[[Image:PBL6.gif]]
* R2 U' B2 U2' R2' U' R2
* R2 U' F2 U2' R2' U' F2
* R2 U R2 U2 F2 U F2
* R2' U R2 U2 y' R2' U R2
* L2' U F2' U2 R2' U R2'
</td>

</tr>
</table>

The first two algorithms (Y perm and A perm) only affect the top layer, and because of that, they can be used in LBL methods.

Note that D moves are the same as U + y, just a different notation, what you do for real is something in between. The effect of it is that you transform F moves to R or L (D R makes the same turns as U F + y').
== See also ==
* [[Ortega]]
* [[Guimond]]
* [[OLL (2x2x2)]]
* [[PLL (2x2x2)]]

== External links ==
* [http://dtwoner.110mb.com/index.php?p=1_40_PBL David Woner's PBL algs]
* [http://www.cubewhiz.com/ortegapbl.php Bob Burton's PBL algs]
* [https://www.speedcubingtips.eu/pbl-methode-ortega/ PBL algs on Speedcubingtips.eu]

[[Category:2x2x2]]
[[Category:2x2x2 substeps]]
[[Category:2x2x2 speedsolving methods]]
[[Category:Algorithms]]

[[Category:Acronyms]]
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PBL