Great Insights into Rubik type Puzzles
--------------------------------------

(Great though these insights may be, I certainly do not credit myself
 which the original insights except perhaps the one about the super
 flip. Even then it was Mike Reid who wrote the Kociemba-inspired
 program to generate minimal processes, including the superflip process
 shown below. David Benson, a pioneer of computer cubing, found the
 process for generating a sixth move out of the other 5 and later on
 Jerry Bryan confirmed it's minimality. Dan Hoey was the math expert
 who first inspired me to think more about rigour and Group Theory.
 Dr. David Singmaster was the original writer and mathematician who
 published a detailed account of the insights of other mathematicians
 and provided many insights of his own.

 One can extend the concept of monoflips, monotwists and monoswaps
 to the Halpern-Meier Tetrahedron and the Megaminx, and surely these
 are some of the most powerful tools for solving Rubik type puzzles.)

Notation

P   = Process
P'  = Process which is the inverse of P
C_X = Central Reflection
q   = quarter or 1/4 turns

Prime Numbers and Cubes
-----------------------

The number of unsolved states of the 3x3x3 cube is prime, 
e.g. 43,252,003,274,489,856,000 - 1 is prime.

The Greatest Prime factor of the number of permutations of any NxNxN
cube puzzle is the first prime number less than or equal to the
greatest cycle length of like pieces.

Example #1
     For the 3x3x3 rubik's cube the greatest cycle length is a 12
     cycle of edges. Therefore the greatest prime factor is 11.
Example #2
     For the 4x4x4 rubik's cube the greatest cycle length is a 24
     cycle of edges. Therefore the greatest prime factor is 23.

P = sequence which moves all U face pieces to D face
then
P D1 P' = a process which effects the U face only

P = sequence which effects one element of U face leaving all other elements
    in place (can be an edge or corner)
then
P U1 P' = a process which effects the U face only
In cube literature, this type of process is called a monoflip or monotwist.

P = sequence which effects one element of the middle slice leaving all
    other elements in place (must be an edge)
then if the one element is FL
P F2 P` F2 = a process which flips two edges

P = R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3    (12 q)
then
P + (P * C_X) = Superflip or 12-flip in minimal q turns

P = R1 L3 F2 B2 R1 L3
then
P U1 P = D1
I.e. process R1 L3 F2 B2 R1 L3 U1 R1 L3 F2 B2 R1 L3   (17 q, 13 q+h)
     which uses 5 generators to create a sixth in minimal q turns

P = R2 F2 B2 L2
then
P U2 P = D2
I.e. process R2 F2 B2 L2 U2 R2 F2 B2 L2 (18q, 9 q+h)
     which uses 5 generators squared to create a sixth squared

Each element of the Rubik group has 12 neighbours in the q metric.
Each element of the Megaminx group has 24 neigbours.

Composed and collated by Mark Longridge, Sept. 14, 1997