Orbit Solver Notes
O1 – Overview
Each orbit is solved *separately*: any solution algorithm will *not* move stickers located outside the current orbit.
O2 – Features and Limitations
The Solver will provide complete solution algorithms only if:
1. 'Initial' and 'Goal' permutations are those of legal cube states
2. 'Delta' permutation *total* parity of a given orbit is even:
• Corners: permutation parity even + sum of all twists equal to zero.
• Midges: permutation parity even + sum of all flips equal to zero.
• Edges: permutation parity even.
• Centers: permutation parity even.
So, it's up to the user to provide strings of cycles that will fill these conditions.
Checking the parity of a permutation can be achieved by counting cycles involving an even number of letters each and verifying that the count is indeed even.
Example: (DAB)(CIKF)(PG) is a permutation of even parity because there are 2 cycles (CIKF) and (PG) of even length.
Notice that 'Initial' and 'Goal' permutations must be of the same parity, either even or odd, and not necessarily even.
O3 – Permutation Cycle Notation
Permutation cycle notation: cycles of letters must be enclosed in parenthesis, disjoint and with no repeated letters.
• Corners: a single letter per corner piece is sufficient as the Solver will complete the permutation by adding the 2 missing letters.
There are 3 letters per corner and any of these 3 letters is allowed to appear only *once* in the cycle notation, provided that *all* permuted corners are distinct.
All 8 groups of 3 letters are given below for convenience:
(D,O,F), (A,E,I), (B,L,S), (C,R,P), (W,J,H), (X,G,N), (U,M,Q), (V,T,K).
Example: (DJAG)(BMCK), where corner (D,O,F) is represented by letter 'D' so that there is no longer need to use missing letters 'O' and 'F' in cycles.
Notice that even though letter 'D' appears only once in the first cycle, then letters 'O' or 'F' can't no longer appear in subsequent cycles, to ensure that permuted corners are indeed distinct.
• Midges: a single letter per midge piece is sufficient as the Solver will complete the permutation by adding the missing letter.
There are 2 letters per midge and any of these 2 letters is allowed to appear only *once* in the cycle notation, provided that *all* permuted midges are distinct.
All 12 groups of 2 letters are given below for convenience:
(I,H), (S,K), (P,Q), (F,N), (E,D), (W,G), (T,U), (B,R), (A,L), (O,C), (X,M), (J,V).
Example: (DAB)(CIK)(FPG)(MTJ), where midge (E,D) is represented by letter 'D' so that there is no longer need to use missing letter 'E' in cycles.
Notice that even though letter 'D' appears only once in the first cycle, then letter 'E' can't no longer appear in subsequent cycles, to ensure that permuted midges are indeed distinct.
O4 – Corner Twists
• Corner twists: corners can be twisted *inplace*, provided the sum of all twists is zero.
Example: (DOF)(AIE)(CPR)(BLS), where 4 corner are twisted inplace.
• Cycles of corners + corner twists are allowed in a same string, provided all cycles are disjoint.
Example: (DJAGB)(CPR)(MQU), where the first 5 corners are permuted and the 2 other twisted.
• Twisted corners 3, 5 or 7cycle are allowed in a same string, provided *all* corner stickers are listed.
Example: (BCGLRNSPX)(MUQ), where the 3 corners (B,L,S), (C,R,P) and (G,N,X) are permuted, with the sum of their 3 twists != 0 and a fourth corner (M,U,Q) is twisted inplace to set the sum of all 4 twists to zero again.
O5 – Edge Flips
• Midge flips: midges can be flipped *inplace*, provided the sum of all flips is zero.
Example: (DE)(BR)(AL)(CO), where 4 midges are twisted inplace.
• Cycles of midges + midge flips are allowed in a same string, provided all cycles are disjoint.
Example: (DABCI)(KS)(FN), where the first 5 midges are permuted and the 2 other flipped.
• Flipped midges 3, 5, 7, 9 or 11cycle are allowed in a same string, provided *all* midge stickers are listed.
Example: (AHCLIO)(FN), where 3 midges (A,L), (H,I) and (C,O) are permuted, with the sum of their 3 flips != 0 and a fourth midge (F,N) is flipped inplace to set the sum of all 4 flips to zero again.
O6 – Isograms
Isograms: an isogram is a word or phrase in which no letter is used more than once.
Disjoint cycles can be thought of as isograms, many of which can be displayed on a cube as strings of permuted stickers, mainly on edges or centers though.
Examples: (RUBIK), (VOID)(CUBE), (WORDGAMES), (ALGORITHM), (WORKMANSHIP), (BUCKFASTLEIGH), (DUBROVNIK), (FOX)(DELTA), (GOLF)(MIKE)
Copyright © 2011
André Boulouard, Walter Randelshofer, Werner Randelshofer
All rights reserved.
