Pyraminx Crystal - Last Layer Algorithms

Here is one method for solving the last layer of the Pyraminx Crystal. I hope that it will be of some interest to fans of this puzzle, both speedsolvers and leisurely solvers -- anyone who enjoys comparing methods and solving as efficiently as they can. Suggestions, corrections, improvements, and alternatives are welcome and encouraged.

Overview

Stage 1. Orient corners.

Stage 2. Permute corners.

Stage 3. Lower edges. Includes fixing the orientation of moved edges - ignoring any lower edges that are correctly positioned but flipped.

Stage 4. Permute upper edges (ignoring their orientation).

Stage 5. Flip edges.

Average Turn Count (face turns / fifth turns):

Stage 1. 9.01/9.57

Stage 2. 12.32/13.27

Stage 3. 28.91/31.69

Stage 4. 7.52/8.00

Stage 5. 10.96/18.41

Total. 68.72/80.94

Notation

L, F, R, BR, B, BL form a zigzag equator around the puzzle (e.g. Dark Blue, Pink, Green, Blue, Purple, Dark Green). Clustered around the north pole are UL, U, UR (Red, Yellow, Orange); and clustered around the south pole are DL, D, DR (Pale Blue, White, Brown). Rotating the puzzle is indicated in lower case within parentheses, e.g. (r) = clockwise rotation around the R face.

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Stage 1. Orient corners. Each case lists the corners in clockwise order around the U face from UFL to UFR, where A means the corner needs twirling anticlockwise, C means the corner needs twirling clockwise, and - means the corner is already correctly oriented. The 16 cases all have 5/81 chance of occurring, with 1/81 chance of skipping this stage.

2 corners:

[---CA] F U R U' R' F'

[C---A] (r') R U R' DR R U' R' DR'

[C-A--] F R U R' U' F'

[C--A-] (r') DR R U R' DR' R U' R'

3 corners:

[--CCC] R U R' U R U2' R'

[A--AA] R' U' R U' R' U2 R

[-C-CC] R U' R' F R' F' R or [CC-C-] R' U2' R U R' U R

[A-AA-] R' F R F' R U R' or [-AA-A] R U2 R' U' R U' R'

4 corners:

[A-CCA] F U R U' R' U R U' R' F'

[C-AAC] F R U R' U' R U R' U' F'

[CA-CA] R' U' R U' R' U R U' R' U2 R

[-ACAC] R' U R U R' U R U' R' U' R

[-CAAC] R U' R' U' R U R' U R U R'

[CA-AC] R' U R U R' U' R U' R' U' R

5 corners:

[CCCAC] R U R' U R U2' R2' U2' R U R' U R

[AAAAC] R' U' R U' R' U2 R2 U2 R' U' R U' R'

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Stage 2. Permute corners. The corners are numbered from 1 to 5 in clockwise order around the U face from UFL to UFR. Probabilities: F, T = 5/12 each; U = 1/12; skip stage = 1/12. (A twist of the U face to fix the corners is a "skip", even though it contributes an average of 0.8/1.2 to the turn count when it occurs.)

[3,4,5] (F1 -- Clockwise adjacent 3-cycle)

R' U2' R U R' U2 L U' R U L'

[3,5,4] (F2 -- Anti-clockwise adjacent 3-cycle)

L U' R' U L' U2' R U' R' U2 R

[2,4,5] (T1 -- Non-adjacent clockwise 3-cycle)

R' U' F' U F R (u2 r') R' DR R U' R' DR' R U

[2,5,4] (T2 -- Non-adjacent anti-clockwise 3-cycle)

(u) F U R U' R' F' (u2 r') R U' R' DR R U R' DR'

[1,3] [4,5] (U -- Unequal length pair swaps*)

(r') R U' R' DR R U R' DR' R U R' DR R U' R' DR'

* Megaminx algo from Erik Akkersdijk's website (

http://erikku.er.funpic.org/rubik/).

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Stage 3. Lower edges. Includes fixing the orientation of moved edges - ignoring any lower edges that are correctly positioned but flipped. For this stage and the next, F is the last face. The following algorithms move an edge to R/DR. The only edges affected are R/DR, the source edge, and one other (upper) final face edge, so we can solve the lower 5 edges in any order we like. [DR/DL] means that the DR/DL edge's DR tile colour matches the R face, and its DL tile colour matches the DR face.

[DR/DL] (f') R DR U DR' U' R'

[DL/DR] (dl') U' R' DR U' DR' U R U

[DL/L] R U2 BL' U' BL U' R'

[L/DL] (r) L2 U' DR U DR' L2'

[L/U] (r) UL L DR' L DR L2' UL'

[U/L] (r) UR2' U DR' U' DR UR2

[U/R] R U DR U' DR' R'

[R/U] (dr') DR R U' DR U DR' R' DR'

[F/DR] R DR' L DR L' R'

[DR/F] DL' U' BR U' BR' U2 DL

[F/DL] R DR U' DR' U R'

[DL/F] (r) DR R2' F UR' F' UR R2 DR'

[F/L] R U' DR U DR' R'

[L/F] (f') R' DR U' DR' U R

[F/U] DR2' DL' R L' R' L DL DR2

[U/F] DR' L' R L R' DR

[F/R] U DL BR' DL BR DL2' U'

[R/F] DR' R L' R' L DR

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Stage 4. Permute upper edges (ignoring their orientation). The edges are numbered 1 to 5 clockwise from F/DL to F/DR. My names for the upper edge perms are

Flat triangle,

Tall triangle,

Equal pairs,

Unequal pairs,

X,

B, figure of ei

Ght,

Pentagon, and

Star. Probabilities: F, T, B, G = 1/6 each; E, U, X = 1/12 each; P, S = 1/30 each; skip stage = 1/60.

F1 [1,2,5] L' R2 (r') L' R' L R' (r) L

F2 [1,5,2] L' R (r') L' R L R2' (r) L

T1 [1,3,5] L' R L R'

T2 [1,5,3] R L' R' L

E [1,5] [2,3] L' R L R2' (r) L R L'

U [1,2] [3,5] L' R L R' (r) L R2' (r) L R L' R (r') L'

X [1,3] [2,5] T1 + F1

B1 [1,3,4,2,5] L' R L R' (f) R L' R' L

B2 [1,5,2,4,3] (f) L' R L R' (f') R L' R' L

G1 [1,3,2,4,5] L' R L R' (f) L' R L R'

G2 [1,5,4,2,3] (f) R L' R' L (f') R L' R' L

P1 [1,2,3,4,5] L' R L R2' (r) L R L2' (l) R L R'

P2 [1,5,4,3,2] R L' R' L2 (l') R' L' R2 (r') L' R' L

S1 [1,3,5,2,4] L' R L R' (f') L' R L R'

S2 [1,4,2,5,3] R L' R' L (f) R L' R' L

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Stage 5. Flip edges. This stage takes an average of 11 face turns. Sometimes a setup move is needed with the first algo. The most common situation, occurring 53.7% of the time, is 2 flipped upper edges and no flipped lower edges. It's very rare (<0.01%) to have 4 flipped lower edges, but for completeness the last algo takes care of this.

Flip F/L and F/R:

DR2' D2 DR2' U' DR2 D2' DR2 U

Flip F/L and F/DL:

DR' D2 DR2' U' DR2 D2' DR2 U DR'

Flip F/R, F/L, L/UL, and UL/UR (ring of 4 lower edges):

DL DR2' DL2' DR U2' DR' DL2 DR2 DL' U2

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[Edit - fixed a typo in the first Stage 3 algo.]