From:

~~~ ~~~ Subject:

Something new to stop the drought of cube posts...

Example of Invariant Shifting -----------------------------

The resultant position generated by process p8 is invariant under

shifting, specifically 2 X on the Left and Right sides.

P8 2 x ORDER 2: shift 0 D2 F2 T2 F2 B2 T2 F2 T2 1 T2 D2 F2 T2 F2 B2 T2 F2 2 F2 T2 D2 F2 T2 F2 B2 T2 3 T2 F2 T2 D2 F2 T2 F2 B2 4 B2 T2 F2 T2 D2 F2 T2 F2 5 F2 B2 T2 F2 T2 D2 F2 T2 6 T2 F2 B2 T2 F2 T2 D2 F2 7 F2 T2 F2 B2 T2 F2 T2 D2

This is the longest process I've found so far. Certainly this property

is not true of all squares group processes. I suspect there are no

processes in the full group with this property (of any significant

length). Perhaps the fact that the L and R faces never rotate will

give some clue on how to generate processes with this property.

Q: Is this the longest such process?

Further Notes on Antipodes in the Square's Group ------------------------------------------------

I just realized some things about sq group antipodes which

I should have seen before...

The closest 2 antipodes can be is 2 square's group moves.

Take the position produced by p66:

p66 Double 4 corner sw L2 B2 R2 F2 L2 F2 T2 R2 (T2 D2 F2 T2) F2 L2 D2

Any turn will reduce this to a position requiring 14 moves. Undoing

this move will regenerate the antipode. No single move can change

position p66 into another antipode, therefore the closest any 2

antipodes can be is 2 moves.

Futhermore any antipode can not be made into a local maximum

which is 14 moves deep with 1 half turn. I will conclude that

there are no local maxima in the square's group that

neighbour each other closer than 2 moves.

-> Mark <-

Email: mark.longridge@canrem.com