PuzzleSolver.com
  Home > Sequential > Rubik's Cube > Puzzle of the Month    Puzzle Store   
 Contents
  Terms to know
  Step 1
  Step 2
  Step 3
  Step 4
  Step 5
  Step 6
  Step 7
  Step 8
  Patterns
  Take it apart
  Java Cube
  Advanced Moves
  Cube History
  Cube Links

 


 For Sale At

Puzzle Master

Amazon.com

Cubezz

 Similar Puzzles

similar puzzle

similar puzzle

similar puzzle

similar puzzle


Advanced Moves for Rubik's Cubes

Compiled and maintained by Mark Longridge


Use the following sequences generate the described pattern from a solved state, and it's inverse to get back from the pattern to a solved state.

In general patterns with T for Top are my own discoveries, while ones containing U for Up are the work of others. The contributions of Mike Reid, Dan Hoey, Dik Winter and Hans Kloosterman have been invaluable.

The number of moves of a process is (N), which may be suffixed by s to indicate slice moves. After () L indicates that this process is a local maximum. After () S indicates that this process is shift invariant

Some alternative sequences result in a pattern rotated with respect to the original sequence.


s1 2x2x2 antipode (F1 R1) ^2 + F1 D2 F1 D3 F1 D2 R2 (11)
(note this uses " f, r, d " group only)

p1 6 X of order 3 R2 L3 D1 F2 R3 D3 F1 B3 U1 D3 F1 L1 D2 F3 R1 L2 (16)
(rotates edges 120 degrees around the FTR corner and BDL corner, 20q)
p1a alternate method 1 U3 D1 R2 L2 F1 U2 D2 F2 R2 L2 F1 L3 F2 B2 R2 U2 (13s)
D2 L3 U1 D3
p1b alternate method 2 (Mr2 D3 Mr2 U1) ^3 TOP becomes LEFT (28s)
(Mr2 D3 Mr2 U1) ^3 LEFT becomes TOP
Mr3 Mt1 Mr1 Mt3
p2 6 X of order 6 F1 R1 L3 U2 F2 B2 U3 D3 F1 B3 L3 U1 R2 L2 U2 F2 (18)
B2 D1
p3 12 flip R1 L1 D2 B3 L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 (20)S
R2 U3 F2 D3 (28 q moves)
p3a alternate method ( (Mr1 U1) ^4, ^^^, >>> ) ^3 (24s)
Rather cryptic but: ^^^ means TOP face becomes BACK face
>>> means FRONT face becomes RIGHT face
p3b 2 flip (ub, ur) B3 U2 B2 U1 B3 U3 B3 U2 F1 R1 B1 R3 F3 (13f, 16q)
p4 8 flip (Op sides) (R1 L1 T1 D1 F1 B1) ^2 (12)
(element of the anti-slice group)
p5 4 flip (Mid. edges) T3 B2 R3 L1 T2 B3 T2 R1 L3 B2 R2 L2 T1 R2 L2 T1 (16)
p6 12 flip, 8 twist D1 F2 U3 B2 R2 B2 R2 L1 B3 D3 F1 D2 F1 B2 U1 F3 (20)
R1 L1 U2 F3
p7 Cube within a cube R1 F2 R3 U3 R2 U2 R3 U1 F1 D1 B1 R3 B3 D3 U2 F3 (19)
R1 F2 R3
p7a Improvement on p7 U2 F2 R2 U3 L2 D1 (B1 R3) ^3 + D3 L2 U3 (15)
(rotates ring surrounding URF 2x2x2 block 120 degrees counter-clockwise)
p7b Symmetric Maneuver (R3 U1 F2 U3 F3 L1 F2 L3 F1 R1 C_X ) ^2 (20f, 24q)
p7c "u, f, r" only U2 F1 R3 F2 R1 F2 R1 F2 R3 F1 U2 F2 R1 U3 R2 U1
R2 U1 R2 U3 R1 F2 (22f, 32q)
p8 2 X D2 F2 T2 F2 B2 T2 F2 T2 (8)S
p9 4 X (5 generators) T2 B2 T2 L2 R2 T2 F2 T2 (8)
p9a 4 X (4 generators) R2 T2 L2 R2 T2 R2 D2 T2 (8)
p10 6 X order 2 T2 D2 F2 B2 L2 R2 (6)L
p10a alternate method (L3 R1 U3 D1)^3 (12)
(element of the slice group)
p11 2 Line, 2 DOT, 2X L2 R2 B2 T2 D2 F2 (6)
p12 4 DOT, 2X T2 L2 R2 F2 B2 T2 (6)
p13 4 DOT D1 U3 L2 R2 D1 U3 F2 B2 (8)
(element of the slice group)
p13s sq group method L2 R2 D2 L2 B2 L2 R2 F2 R2 D2 (10)
p14 6 Cross order 2 F1 T2 D2 F3 B3 R3 L3 T2 D2 R1 L1 T2 D2 B1 (14f, 20q)
p14a alternate method R3 T2 L2 R2 F2 B2 D2 L2 R2 F2 B2 R1 (12f, 22q)
p15 4 Cross order 4 F2 B2 U1 R2 L2 U2 F2 B2 D3 R2 L2 D2 (12)
p15a alternate method (Mr2 D1 Mr2 T3) ^3 (18)
(rotated from left to right)
p16 4 Cross order 2 L2 R2 B2 L2 R2 T2 D2 F2 T2 D2 (10)
p16a alternate method 1 F1 B1 U1 D1 L2 R2 U1 D1 F1 B1 U2 D2 (12)
(p16a is shorter in the qtw metric being 16 qtw, in anti-slice group)
p16b alternate method 2 (F2 B2 T1 D1 L2 R2) ^2 (12)
(P16b is cyclic and easy to remember)
p16c alternate method 3 R2 F2 R2 U3 D1 F2 R2 F2 U1 D3 (10)
(minimal in q and q+h turns)
p17 4 Diagonal (F1 B1 R1 L1) ^3 (12)
p18 4 Diagonal, 2 Cross L2 D2 F2 T2 (F2 L2)^2 + (T2 F2)^2 + R2 F2 (14)L
P18a alternate method (F1 B1 R3 L3) ^3 (12)
p19 4 H order 4 Mr2 U2 Mf2 D1 Mf2 U2 Mr2 D3 (12)
p20 Mark's Pattern 1 R2 U3 R1 D1 F1 B1 R3 L3 U1 D1 F3 U1 F3 U2 D3 B2 (18)
R2 U1
p21 2 H L2 R2 B2 L2 R2 F2 (6)S
p22 2 DOT, 2 Stripe R1 L1 T2 D2 R3 L3 (6)
p23 4 Stripe Type 1 (B2 L2 F2) ^2 (6)
p23a 4 Stripe Type 2 L2 T2 L2 R2 T2 L2 (6)
p24 2 Cross, 2 DOT,2 H D2 (L2 R2 F2) ^2 + T2 (8)
p24a alternate method U1 D1 R1 L1 U2 D2 F2 B2 R1 L1 U1 D1 (12f, 16q)
p25 6 X minus 2 L2 D2 L2 F2 B2 T2 L2 T2 (8)
p26 4 H order 2 Type 1 T2 L2 R2 B2 L2 R2 F2 T2 (8)
p27 4 Cross, 2 X F2 L2 R2 F2 T2 L2 R2 T2 (8)
p28 2 X, 2 H, 2 Diag R2 B2 L2 F2 T2 D2 L2 F2 (8)
p29 2 DOT, 2 ARM, 2 TRI L2 T2 F2 R2 D2 L2 F2 T2 (8)
p30 2 DOT, 2 Cross T2 F2 L2 T2 D2 L2 F2 T2 (8)
p31 2 X, 2 C, 2 ARM R2 T2 F2 T2 L2 D2 F2 T2 (8)
p32 4 X, 2 H L2 T2 D2 L2 F2 B2 (6)
p33 2 X, 2 DOT, 2 Cross L2 R2 B2 L2 T2 D2 L2 F2 (8)
p34 2 X, 4 Line F2 L2 F2 B2 L2 F2 T2 D2 (8)
p35 FD,BD LB,RB swap R2 B2 T2 L2 D2 F2 L2 T2 (8)
p36 Not Sq U2 B1 U2 B3 R2 B1 R2 F1 R2 F3 (10)
p37 2 H, 2 Swap L2 R2 B2 T2 L2 R2 (F2 D2) ^2 (10)
p38 Double 2 Swap T2 B2 T2 F2 L2 D2 F2 L2 D2 L2 (10)
p39 2 X, 2 Swap L2 F2 L2 R2 F2 D2 L2 D2 R2 D2 (10)
p40 2 Cross, 2 ARM B2 L2 R2 T2 F2 D2 F2 T2 R2 L2 (10)
p41 2 DOT, 2 ARM R2 F2 B2 L2 T2 B2 D2 B2 T2 F2 (10)
p42 2 X, 4 Swap D2 F2 D2 B2 L2 T2 F2 R2 T2 R2 (10)
p43 2 X, 2 H, 2 Swap F2 L2 R2 F2 D2 R2 D2 R2 T2 L2 (10)
p44 2 Line, 2 Sw, 2 Arm R2 T2 R2 D2 L2 B2 T2 B2 T2 F2 (10)
p45 4 TRI L2 B2 T2 (L2 F2) ^2 + B2 T2 F2 (10)
p46 4 TRI, 2 Swap L2 B2 T2 L2 F2 L2 T2 F2 T2 D2 (10)
p47 4 Cross, 2 Swap (L2 F2) ^2 + T2 D2 L2 D2 T2 F2 (10)
p48 6 swap B2 L2 F2 T2 R2 T2 R2 (F2 T2) ^2 + L2 (12)
p49 4 DOT, 2 Swap B2 T2 B2 D2 F2 L2 F2 B2 T2 D2 L2 T2 (12)
P50 4 ARM, 2 Swap R2 F2 D2 L2 T2 R2 F2 T2 F2 T2 L2 F2 (12)
p51 2 Sw, 2 TRI, 2 Unit R2 B2 D2 L2 D2 R2 B2 L2 T2 F2 T2 B2 (12)
p52 2 Swap in U Layer L2 F2 D2 L2 D2 F2 L2 B2 T2 F2 T2 B2 (12)
p52a alternate method L1 R1 D2 L3 R3 F3 B3 D2 F1 B1 (10)
p53 4 U, 2 Swap B2 R2 D2 L2 T2 F2 R2 F2 T2 F2 T2 B2 (12)
p54 4 Stump, 2 Swap B2 L2 D2 R2 D2 F2 R2 B2 T2 F2 T2 L2 (12)
p55 6 X minus U Layer R2 F2 D2 L2 D2 F2 L2 F2 T2 F2 T2 B2 (12)
p56 4 Z, 2 ARM F2 D2 B2 L2 T2 L2 D2 F2 T2 F2 T2 L2 (12)
p57 4 Stump A R2 T2 L2 B2 D2 L2 F2 L2 T2 F2 T2 L2 D2 T2 (14)
p58 4 Stump B L2 D2 L2 B2 D2 L2 B2 L2 T2 F2 T2 L2 D2 T2 (14)
p58a alternate method L3 R3 T2 L1 R1 F3 B3 T2 F1 B1 (10)
p59 2 X, 4 U L2 T2 R2 F2 D2 L2 B2 L2 T2 F2 T2 L2 D2 T2 (14)
p60 2 Line, 2 St, 2 U R2 D2 L2 B2 D2 L2 B2 L2 T2 F2 T2 L2 D2 T2 (14)
p61 4 Diagonals R2 T2 B2 D2 (B2 L2) ^2 (T2 F2) ^2 + L2 F2 (14)
p62 2 DOT, 2 Diag, 2 Z L2 T2 B2 D2 F2 R2 D2 R2 T2 F2 T2 F2 L2 T2 (14)
p63 4 Diagonals, 2 DOT R2 D2 F2 T2 B2 L2 D2 R2 T2 F2 T2 F2 L2 T2 (14)
p64 4 Z R2 (D2 B2) ^2 + L2 F2 L2 (T2 F2) ^2 + R2 F2 (14)
p64a 4 Z (antislice) F1 B1 L3 R3 F1 B1 L1 R1 F3 B3 L1 R1 (12)
p65 4 Diagonals, 2 X R2 T2 F2 D2 B2 R2 F2 L2 (T2 F2) ^2 + L2 F2 (14)
p66 Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 T2 D2 F2 T2 F2 L2 D2 (15)
P66a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U1 D1 (13)
(This is an antipode in the squares group!)
p67 Antipode 2 R2 B2 (D2 F2) ^2 + T2 L2 T2 D2 F2 T2 L2 T2 B2 (15)
p67a alternate method F2 R2 F2 U3 D3 L2 B2 D2 L2 B2 U1 D1 B2 (13)
p68 Cherries F1 B1 L3 U3 D3 F3 B2 D2 R2 B1 R3 L3 U3 R2 B2 U1 (18)
D2 F2
p69 Tetra Peak exch U3 F2 L2 B2 D2 F2 R3 U1 D1 F3 R3 L1 F3 B1 R1 U3 (16)
p70 Fraser's Pattern T2 L2 F2 L2 B2 L2 T2 F2 L2 T2 (10)
p71 4 DOT, 2 H R2 F2 T2 D2 F2 T2 D2 L2 T2 D2 (10)
P72 2 X, 4 H L2 T2 D2 L2 F2 T2 D2 F2 T2 D2 (10)
p73 6 Cycle of corners L2 F2 T2 F2 T2 L2 T2 F2 T2 F2 (10)
p74 3 M Pattern L2 B2 T2 B2 L2 T2 L2 D2 T2 F2 (10)
p74 4 Stump C D2 F2 T2 B2 R2 B2 T2 R2 T2 F2 L2 F2 (12)
p75 4 Corner Op Swap T2 F2 R2 F2 T2 R2 T2 F2 T2 F2 T2 F2 R2 (13)
p76 4 Diag Corner Swap L2 D2 B2 D2 B2 R2 B2 L2 T2 F2 T2 F2 R2 F2 (14)
p77 4 Opp Corner Swap T2 F2 R2 F2 T2 R2 T2 F2 T2 F2 T2 F2 R2 (13)
p77a alternate method F2 R2 U1 D1 F2 L2 U1 D1 B2 L2 U2 (11)
p78 2 sided Oh's T2 B2 L2 B2 T2 L2 T2 F2 T2 F2 T2 F2 L2 (13)
p79 2 Four's F2 L2 R2 D2 F2 L2 F2 B2 D2 F2 T2 L2 (12)
p80 2 DOT, Invert T's R2 B2 D2 R2 (B2 L2)^2 + T2 D2 F2 T2 F2 L2 T2 (15)
p80a alternate method U1 F2 R2 L2 U2 D2 F2 U2 D3 (9)
p81 Snake type 1 B1 R1 L3 D3 R2 D1 R3 L1 B3 R2 U1 B2 U3 D1 R2 D3 (16)
p82 8 Crn Twist 1 F1 R2 L2 U2 D2 F2 B3 R2 U1 F2 B2 R2 L2 U2 D3 L2 (16)
p83 4 DOT + 4 Diagonal R2 D2 B2 T2 B2 R2 B2 L2 (T2 F2) ^2 + L2 F2 (14)
p83a alternate method D3 F2 R1 U2 F2 B2 D2 R2 L3 F2 U3 D2 (12)
p84 2 DOT, 2 T, 2 C T2 L2 T2 F2 R2 B2 T2 L2 F2 T2 F2 T2 F2 (13)
p85 2 DOT, 2 sided C's D2 B2 R2 F2 T2 L2 T2 F2 T2 F2 T2 F2 L2 (13)
p86 Twisted Rings B1 R1 B3 R2 U3 F3 L1 U1 R1 D1 L1 U3 B3 L1 U1 L2 (16)
p87 Twisted Cube Edges D1 L3 B1 R1 D3 R3 D1 B3 L1 B1 R3 B3 R1 D3 (14)
p88 4 Stump D R2 F2 R2 D2 R2 F2 L2 T2 (8)
p88a alternate method L3 U2 R1 L3 B2 R1 (6)
p89 2 Crss, 2tri, 2bar F2 B2 D2 L2 B2 T2 L2 T2 F2 L2 (10)
p90 2 Crss, 2 H, 2 ARM T2 R2 F2 T2 D2 F2 T2 R2 T2 L2 (10)
p91 4 Diagonal, 2 Line L2 D2 B2 T2 B2 L2 F2 L2 T2 F2 T2 F2 R2 F2 (14)
p92 4 Stripe (Full) R1 U1 D1 R2 F1 U3 D1 F3 B1 U3 R3 L3 B1 L2 U1 D3 (17)
B2
p93 Mark's Pattern 2 U1 R3 B3 U3 F2 B2 D1 B1 L3 D1 F2 R2 F2 U3 F2 D3 (16)
p94 Kociemba Ph2 Max L2 U1 R2 B2 U2 B2 L2 D2 L2 F2 D3 L2 B2 F2 L2 F2 (18)
U3 D1
p95 Ph2 Max + 12 flip F2 D1 R1 L1 B1 D1 U3 F3 B1 D3 F3 B3 R1 F1 B3 U2 (20)
R2 L2 F2 R2
p96 3 TRI, MIDDON R2 T2 R2 F2 D2 F2 L2 B2 F2 T2 F2 L2 (12)
p97 6 T's A B2 R2 T2 R2 D2 B2 T2 L2 T2 L2 (10)
p98 6 T's B D2 B2 L2 F2 T2 B2 T2 L2 T2 B2 (10)
p99 2 DOT, 4 ARM R2 B2 D2 L2 B2 L2 F2 L2 T2 D2 F2 T2 F2 L2 T2 (15)
p99a alternate method U1 R2 F2 B2 U2 D2 R2 D1 (8)
p100 2 Cross, 4 ARCH R2 B2 T2 R2 F2 L2 F2 L2 T2 D2 F2 T2 F2 L2 T2 (15)
P100a alternate method F2 U2 D2 F2 R2 L2 D1 F2 R2 L2 B2 U1 (12)
p101 6 Cross Order 3 F1 L2 R2 F1 B1 L2 R2 F1 L1 B2 F2 L1 R1 B2 F2 L1 (18)
(Plummer's Cross) U2 D2 (28 q moves)
p102 2 Di, 4 T (1) B2 T2 R2 B2 T2 R2 F2 T2 L2 (9)
p103 2 Di, 4 T (2) D2 F2 R2 D2 F2 R2 T2 F2 L2 (9)
p104 4 T, 2 ARM F2 D2 F2 L2 T2 L2 F2 T2 F2 (9)
p105 2 Di, 2 L, 2 C R2 D2 R2 B2 T2 L2 T2 R2 T2 F2 D2 (11)
p106 2 Cross, 4 Z dis L2 D2 B2 D2 B2 R2 F2 L2 T2 F2 T2 F2 R2 F2 (14)
p107 4 T, 2 Z T2 R2 F2 T2 L2 B2 T2 L2 F2 (9)
p108 2 DOT, 2 T, 2 ARM L2 F2 T2 R2 B2 L2 F2 L2 T2 D2 F2 T2 F2 L2 T2 (15)
p108a alternate method R2 F2 B2 L2 D1 R2 U2 R2 L2 U2 R2 D1 (12)
p109 2 DOT, 4 T D2 B2 L2 F2 T2 R2 D2 F2 T2 F2 T2 F2 L2 (13)
p110 6 Small C L2 B2 T2 F2 T2 L2 T2 F2 T2 F2 (10)
p111 2 DOT, 4 Z (SOS) L2 T2 F2 D2 F2 R2 B2 L2 T2 F2 T2 F2 R2 F2 (14)
p112 4 U order 4 F1 R3 B3 D2 L3 U1 D3 B1 D2 R1 F1 L3 (12)
(compare with page 135 of Rubik's Cubic Compendium = 25)
p113 4 Cross O2 (Full) F1 R2 L2 U2 F2 B2 R2 L2 D2 F1 B2 (11)
(This is also known as Christman's Cross = 20 q moves)
p113a alternate method F1 U1 D1 L2 R2 U1 D1 F1 B1 U2 D2 B1 (12)
(Christman's Cross = 16 q moves)
p114 3 ARM Order 4 F1 B3 L1 F2 L2 U1 D3 B1 U2 R1 L3 B2 D1 L2 U3 (15)
p115 RCC 3.1.14 F1 R1 U1 R3 U3 R1 U1 R3 U3 R1 U1 R3 U3 F3 U1 (15)
p116 Improve RCC 3.1.14 F1 R2 F2 U2 F1 R2 F3 U2 F2 R2 F3 U1 (12)
p117 Improve RCC 3.9.36 U1 F3 D3 L1 B2 L3 D1 F1 D1 L2 U3 D3 F2 D3 F2 L2 (16)
p118 4 T, 2 Cross L2 B2 D2 R2 D2 F2 L2 B2 T2 F2 T2 F2 T2 (13)
P119 Mark's Pattern 3 F2 U2 D2 F1 R3 D1 R2 D3 R1 F3 U1 R2 U2 R2 D2 F2 (16)
p120 6 Stripe (1) R1 F1 B1 D1 R3 L1 U3 D1 L3 U3 D3 B1 D2 L2 D2 B2 (17)
D1
p121 Improve RCC 3.9.31 R1 D2 F1 R3 L1 U1 D3 L1 F3 B1 D3 L3 B2 U3 (14)
p122 Mark's Pattern 4 U3 B3 R3 U1 F3 U1 D3 L1 U3 F1 R3 U1 R2 L2 D3 L2 (17)
D1
p123 Cube in a cube 2 F2 R2 U2 F2 R2 U3 R2 D2 B2 U3 L2 U2 L2 B2 U1 (15)
p124 Improve RCC 3.9.32 U1 R2 U3 L1 D2 L3 F1 U2 F3 U1 L2 D2 R2 L2 U1 (15)
p125 ML's Multicolour 1 U1 B1 U3 B1 D2 L2 U3 F3 D1 B1 R1 L3 U1 D1 F2 U2 (18)
D2 L2
p126 ML's Multicolour 2 U2 F1 R2 F2 U2 B2 R2 U2 B2 D2 B3 U3 D3 F2 U1 D3 (16)
p127 2 Small Snakes B2 R2 F2 D2 B2 L2 D2 F2 T2 F2 (10)
p128 2 H, 2 T, 4 corner L2 B2 R2 F2 L2 F2 T2 R2 T2 D2 F2 T2 F2 L2 T2 (15)
p128a alternate method F2 R2 U2 F2 R2 U3 D3 B2 L2 F2 B2 U3 D3 (13)
p129 2 H, 2 T, 2 ARCH R2 F2 L2 F2 L2 F2 T2 R2 T2 D2 F2 T2 F2 L2 T2 (15)
p130 2 Cross, 4 ARCH 2 L2 B2 D2 B2 L2 D2 F2 L2 T2 D2 F2 L2 F2 L2 T2 (15)
p130a alternate method F2 R2 F2 B2 U1 D1 F2 R2 D2 F2 L2 U3 D3 (13)
p131 2 H, 2 ARM, 2 ARCH L2 F2 R2 D2 B2 L2 D2 L2 T2 D2 F2 T2 F2 L2 F2 (15)
p132 2 Cross,2 arch,2crn L2 F2 D2 R2 F2 L2 B2 L2 T2 D2 F2 T2 F2 L2 D2 (15)
p133 2 Cross, 2 T, 2 ARM L2 F2 D2 F2 D2 F2 T2 L2 T2 D2 F2 T2 L2 T2 B2 (15)
p133a alternate method R2 U1 F2 R2 L2 U2 D2 F2 U2 D3 R2 (11)
p134 2 CRN, 2 X, 2 ARCH L2 F2 D2 B2 T2 F2 T2 L2 T2 D2 F2 T2 L2 T2 B2 (15)
p135 2 X, 4 T L2 B2 D2 F2 T2 F2 T2 L2 T2 D2 F2 T2 L2 D2 F2 (15)
p135a alternate method F2 R2 F2 U3 D3 L2 B2 D2 R2 B2 U1 D1 F2 (13)
p136 2 H, 2 ARM, 2 CRN R2 F2 L2 T2 B2 L2 T2 L2 T2 D2 F2 T2 F2 L2 B2 (15)
p136a alternate method U2 F2 R2 L2 D1 F2 R2 L2 B2 U1 B2 D2 (12)
p137 2 X, 4 ARM L2 F2 T2 B2 T2 F2 T2 L2 T2 D2 F2 T2 L2 D2 F2 (15)
p137a alternate method R2 U1 R2 F2 B2 U2 D2 R2 D1 L2 (10)
p138 Mark's Pattern 5 R1 L3 D2 F1 R2 U3 D1 B2 R1 F3 B1 D2 L2 U1 F2 B2 (17)
U3
p139 6 X + 12 flip F1 B3 R1 U1 D3 B2 R2 B3 R3 L1 F2 B2 U3 F2 D3 F2 (20)
B2 D3 R2 D3 (28 q)
p139a alternate method B3 L1 D1 L1 F3 U3 D3 L1 B3 D3 F3 R1 L3 F3 U1 D1 (20)
L2 U1 D1 B2 (22 q)
p141 Superfliptwst + 6X F1 R3 U3 D1 R3 B1 D3 F2 R2 L1 F1 R3 B3 L1 D2 B2 (22)
U3 R2 L2 D3 R2 L2
p141a alternate method F1 R1 L2 U3 R2 L3 U3 D2 R2 F1 D1 B1 D1 F2 U3 R3 (20)
D3 F2 D2 L2
(**This process was one of the hardest ever to reduce to 20 moves, requiring
over 19 hours on an SGI R4K Indigo, 28 q turns, 16+4**)
p142 Pair corner, edges F1 L2 F2 R1 L3 D3 L1 D3 R1 D1 F3 R2 U3 B2 U1 F2 (17)
(markp4.q) L2
p143 Pinwheels F1 B1 L1 R1 F3 B3 U3 D3 L1 R1 U1 D1 (12)
(Another antislice pattern, 4 M-conjugates)
p144 Mark's Pattern 7 R3 U1 F1 U3 F2 B1 D1 R3 F1 L1 D3 B1 U2 D2 F2 U1 (16)
p145 Improve RCC 3.4.24 F3 B2 U1 R3 U3 F1 U1 R3 F3 R3 F1 R2 U3 B2 (14)
p146 Mark's Pattern 9 F2 B2 R1 L3 B3 U1 D3 R2 F2 R3 F1 B3 D1 B2 U3 (15)
p147 ML's 6 ARM Pattern U3 F3 L3 U3 F2 B2 D1 R1 F3 B2 D2 R2 U2 D2 R2 U3 (16)
p148 C14 or RCC 3.2.32 B2 L2 U1 F3 B3 R1 F3 B1 U3 L2 B2 D1 R2 (13)
p149 B3 or RCC 3.2.6 F3 R3 D3 R1 F1 R3 L2 U1 B1 U3 R1 L2 (12)
p150 C12 or RCC 3.2.27 R1 D1 R3 F3 U3 R2 U1 R1 F1 D3 R2 D2 R1 (13)
p151 C3 or RCC 3.2.25 R3 D2 R2 D1 F3 R3 U3 R2 U1 F1 R1 D3 R3 (13)
p152 Antwist p66 + p82 R1 F2 B2 D2 R1 L3 B2 R1 B2 U1 F2 U2 F2 D2 F2 R2 (18)
L2 D3
p153 ML's Checkerboard B1 U2 R1 L1 D2 B3 L2 F2 R1 F3 U3 D3 F3 B3 R2 U1 (19)
(Mark's Pattern 12) R2 D3 L2
p154 (FRBL)^5 Pattern F1 L3 D2 F3 B2 R1 L3 F2 B3 R2 B1 U2 D2 R3 D2 L2 (19)
B2 L2 F2
p155 8 Crn Twist 2 U1 F2 L1 F2 B2 U2 D2 R3 L2 B2 U2 D3 F2 B2 R2 L2 (16)
p156 8 Crn Twist 3 (L1 R2 F2 B3) ^4 (16)
P157 8 Crn Twist 4 (U2 R2 L3 D1) ^4 (16)
p158 Mark's Pattern 14 U3 R2 U1 R2 B3 L3 U3 R2 L2 D1 L1 F3 R2 U3 B2 D1 (16)
p159 6 X + p35 F2 B2 D2 L2 B2 D2 L2 T2 F2 L2 (10)
p160 4 H order 2 Type 2 B2 D2 L2 R2 F2 L2 R2 F2 T2 F2 (10)L
P160a alternate method (F2 B2 R2 L2 T2) ^2 (10)
p161 Mark's Pattern 16 (F1 R1 L1 B1) ^3 + F2 B2 D2 F2 B2 T2 (18)
p162 2 X, 4 H full (F1 T2 B1) ^4 (12)
p163 4 ARM Full (F2 T1 B2) ^4 + T1 D3 (14)
p164 4 Y's Rotated (F1 T2 D2) ^6 + F1 (19)
p165 2 Swap, 4 H full (F1 L2 T2 R2 B1) ^2 + L2 R2 T2 D2 (14)
p166 2 H adj swap (F1 L2 T2 R2 B1) ^2 + L2 T2 R2 D2 L2 T2 (16)
p166a alternate method a D2 Mf1 Mr3 D3 Mr1 D2 Mf3 D1 (12)
p166b alternate method b D3 B1 F3 R2 T1 D3 F1 T3 D1 B3 F1 D2 (12f, 14q)
p167 Mark's Pattern 17 (T2 L2 R2 D1) ^2 + D1 (8)
p168 4 H adj swap R2 L2 F2 B2 U1 R2 L2 U2 F2 B2 U3 D2 (12)
p169 4 Y's Rot + 2 X (F2 B2 D1 L2 R2) ^2 + T1 (11)
p170 2 edge swap + 2 H (F2 B2 T1 F2 B2 D1) ^2 (12)
p171 6 Edge Flip (U1 R3 F1 R1) ^5 (20)
(flips uf, ul, ub, ur, fl, fd)
p172 2 Layer Edge Flip Mr2 (D1 B1 F1 R1 L1 U1) ^2 + Mr2 (16)
p173 3 H Pattern U2 R2 B2 D3 F2 B2 U1 F2 R2 U2 (10)
(cycles ld->rd->fd and lu->ru->fu)
p174 6 H order 2 type 1 L2 (F2 T2 D2) ^2 + L2 T2 D2 (10)
p175 6 H order 2 type 2 T2 B2 L2 T2 D2 L2 F2 T2 (8)
(rare example of a pattern with symmetry level 2)
p175a alternate method U3 D3 L3 R3 F2 B2 T2 D2 L3 R3 T1 D1 (12)
(note that this is also 16 q turns)
p176 6 L D2 L2 T2 F2 L2 B2 L2 (F2 T2) ^2 + R2 (12)
p177 Twist U centre 180 (U1 L1 R1 U2 L3 R3) ^2 (12)
p178 2L, 2 Fours, 2 arm (F1 D2 R1 L1 B1) ^3 + F2 B2 D2 F2 B2 T2 (21)
(Symmetry: (TL)(FB)(DR) = 180-degree diagonal rotation)
p179 Early 12-flip F2 B2 L1 F2 D3 U1 R2 B1 L2 R2 F3 L2 D1 U3 B2 L1 (26)S
(predates p3) F1 B1 U1 D1 R1 L1 F1 B1 U1 D1
p180 ML Edge Pattern (F1 B1 R1 D1) ^ 12 (48)
p181 Mark's Pattern 18 F1 R2 L2 B2 U1 F2 R2 L2 B2 D1 F1 (11)
p182 F Layer 4 flip (F1 B1 L1 D1 T1) ^6 (30)
p183 6 Twist (18 q) R1 U3 R2 U3 R1 D3 U3 R1 U3 R3 D2 R3 U3 R1 D3 U3 (16)
p184 UR Antipode 1 F2 U3 D2 L3 D3 R1 U2 B2 R1 B2 R3 D1 L1 D3 R2 U1 (17)
(using all 6 generators) D3 (23 q turns)
p185 TDF-45 (phase 2) F2 R2 U3 B2 U2 R2 U3 B2 D2 F2 L2 U3 F2 R2 D2 (15)
p186 Six flip (D1 R3 F1)^30 (90)
p187 Six flip (D1 R3 F1 R2)^6 (24)
p188 Quad Tri (F1 T1 D1 F1)^4 (16)
p188a Quad Tri Sq R2 F2 T2 F2 L2 D2 R2 F2 T2 L2 F2 L2 (12)
p189 10 Flip (F1 T1 F1 R1 D2)^30 (150)
p190 2 Flip (F1 T1 F1 R1 D1)^45 (225)
p191 2 Flip in face (F1 B2 D1 L1)^18 (72)
p192 4 Flip in face (F1 B1 L1 T1 D1)^6 (30)
P193 10 Flip (B3 T1 L1 B1 L1 R1)^45 (270)
p194 12 Flip (R2 D1 T3 B1 L1)^30 (150)
p195 12 Flip (T1 B3 T1 L1 F3 R3)^6 (36)
p195b 12 Flip (R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 C_X) ^2 (22)
p196 3 Edge Cycle U1 L1 F3 R2 L2 B1 L3 B3 R2 L2 F1 U3 (12)
p197 Pons + 6 DOT F2 B2 T1 D3 F1 B3 L3 R1 T1 D3 (10)
(Element and antipode of the slice group)
p198 2 X, 4 Diag no C L2 D2 F2 D2 B2 L2 F2 L2 T2 F2 T2 F2 R2 F2 (14)
p198a Alternate antisl L1 R1 F1 B1 L3 R3 F3 B3 L1 R1 F1 B1 (12)
p199 2 Cross, 4 Z L3 R3 F3 B3 L3 R3 F1 B1 L1 R1 F1 B1 (12)
p200 6 H + 6 X T2 B2 R2 B2 T2 B2 D2 R2 T2 F2 (10)
(see p175 for the exact 6 H pattern)
p201 Pinwheel + Pons L1 R1 F3 B3 L1 R1 T3 D3 F1 B1 T3 D3 (12)
(The only other antislice pattern with 4 M-conjugates)
p202 6 DOT F1 B3 L1 R3 U1 D3 F1 B3 (8)
p203 2 Cross, 2 Bar F2 R2 F2 B2 T2 D2 L2 F2 T2 D2 (10)
p204 Mark's Pattern 61 (D1 F1 D1 C_X) ^12 (36)
p205 12 Flip C_X (D1 F1 D1 R1 C_X) ^30 (120)
p206 4 cross order 2 F1 B1 U1 D1 L2 R2 U1 D1 F1 B1 (10f, 12q)

Impossible Patterns (possible by disassembly)
-------------------

4 X of order 4 (swap orbit)
6 X order 2 type b (swap orbit) T = blue, D = white, F = orange, R = yellow,
B = red, L = green
6 Diagonal (twist orbit)


Using the 4x4x4 cube we can produce a single exchange of centres
and an exchange of edge pairs, and we can invert a single edge pair.
Thus we can construct all the impossible 3x3x3 patterns except those
involving a twist of a single corner!

Impossible Patterns (possible by peeling off stickers!)
-------------------

6 Cross order 2 (Note this is not the same as p14 above as this pattern
tries for opposite colour crosses on opposite sides)
This pattern may also be created by unscrewing the
centres and exchanging them.

Orders
------

There are 73 different orders
Maximal Order 1260 R1 U2 D3 B1 D3 (5 h+q, 6 q)

Process ( F1 B1 R1 D3 ) ^ 231 = I (Not prime as 11 * 7 * 3 = 231)
Process ( F1 R2 D1 L2 ) ^ 99 = I (Not prime as 11 * 3 * 3 = 99)

Orders of Symmetry
------------------

Possible orders of symmetry:

48 R1 U1
24 L2 U2
16 Mark's Pattern 1 (18 q+h, 22 q)
R2 U3 R1 D1 F1 B1 R3 L3 U1 D1 F3 U1 F3 U2 D3 B2 R2 U1
(Also 7 clockwise + 1 anticlockwise corners!), 6 L
12 2 dot, 2 T, 2 ARM (sq group antipode, see p108)
8 6 Dot (a slice pattern)
6 2 DOT, 4 ARM (sq group antipode, see p99)
4 Pinwheels, Pinwheels + Pons Asinorum
3 4 Dot pattern (slice pattern)
2 6 H pattern type 2, T2 B2 L2 T2 D2 L2 F2 T2
1 Pons Asinorum (6 X order 2) or all edges flipped

The cube centres have 4^5 x 2 = 2048 different permutations
Thus 5 of the 6 centres can be in any of the 4 orientations and
the 6th centre is forced to be correct or turned 180 degrees

In the case of Rubik's Cube 4th Dimension, 4 of the 6 centres
have 4 orientations, 4^4 = 256 different permutations

Note that the checkerboard pattern (6 X) can occur 17 ways:
1 way order 2, 8 ways order 3, 8 ways order 6
Note that the (4 X) can occur 3 ways
Note that the (2 X) can occur 3 ways
.... for a total 23 'X' Patterns, 7 of which are in the square's group
There are also 24 'X' Patterns in the swap orbit

There are a total of 23 cross patterns, 3 of which are in the sq group
Note that the (4 Cross) can occur 9 ways (3 sq group)
Note that the (6 Cross) can occur 14 ways

There are 0 cross patterns in the swap orbit

Six Crosses order 3 occurs in 8 ways
Six DOT occurs in 8 ways

Cycles
------

Rare 11-cycle of edges: ( L2 B1 R1 D3 L3 ) ^ 7 (35)
alternate method: F2 R3 U1 D3 B3 D1 L3 U3 D1 B1 L1 D1 B2 U2 D2 R2 B2 D1 (18)

Identities
----------

Shortest identities are of length 8 and are all instances of the
commutativity of the slice squared group, e.g. (F2 B2 R2 L2) ^2 = I

Inversion, rotation, reflection, and shifting all preserve the
number of clockwise/counterclockwise sign changes between
cyclically adjacent elements.

Centre of the Cube Group
------------------------

The centre of the cube group is the process which commutes with every
other process, the only nontrivial example is the 12-flip

Properties
----------

Local Maxima, Global Maxima, Shift Invariance

Comments on Patterns
--------------------

Note that the cube consists of 8 corners with 3 colour tabs and 12 edges
with 2 colour tabs, e.g. 8 * 3 = 12 * 2 = 24

The cube can be oriented in space 24 different ways
The dodecahedron can be oriented in space 60 different ways

4 DOT is an example of a permutation in the squares group which can be solved
faster using q and h moves

There are positions which have only opposite colours on all six sides which
are NOT in the squares group (See p36)

Colourings
----------

There are 12 ways to 6 colour the Alexander Star (720 / 60) = 12
There are 30 ways to 6 colour the 6 sides of the cube (720 / 24) = 30

Orientations in Space of the Platonic Solids
--------------------------------------------
Orient Num
in Conjugacy
Space Classes

Tetrahedron 12 4 Pyraminx Master, Senior and standard
Cube 24 5 Skewb, Rubik's NxNxN Cubes
Octahedron 24 5 Pyraminx Octahedron, Trajber's Octahedron
Icosahedron 60 5 Impossi-Ball
Dodecahedron 60 5 Megaminx













 Site Info
   Contact Us  |   Terms & Conditions  |   Privacy Policy  |   Puzzle Links  |   Copyright © 2014