# How many 26q* maneuvers are there?

How many 26q* maneuvers are there?

Well, obviously we can't say for sure, as it hasn't yet been proved that the 3 known 26q* positions (which are symmetrically equivalent to each other) are the only 26q* positions. In another thread, Herbert Kociemba mentioned that there are "many" such maneuvers, but he did not attempt to generate them all (for the known 26q* positions).

I note that 26q* refers to a maneuver that is 26 quarter turns long and that is known to be optimal in the quarter turn metric. It may also refer to a position that requires a minimum of 26 quarter turns to solve. 26q (without the asterisk) refers to any maneuver 26 quarter turns long, but isn't necessarily optimal for the position it solves.

We also realize that maneuvers can be rather trivially different from one
another. We could consider that the total number of 26q maneuvers
is 12^{26}, allowing any of the 12 quarter turns at any position
in the sequence without any further restrictions.
Typically, when there are two or more consecutive moves that commute,
we will regard that the order of those particular moves don't matter.
For example, U F B R as being essentially the same as U B F R.
Also, we often consider that a half turn can be done through either two clockwise
quarter turns or two counterclockwise quarter turns.
Choosing one or the other is a rather trivial distinction.

So we can talk about canonical sequences where commuting moves are done in a certain order, and half turns are always done using clockwise moves. We can say U layer turns can't follow D layer turns, R layer turns can't follow L layer turns, and F layer turns can't follow B layer turns.

So how many such canonical 26q* maneuvers are there? For the known 26q* positions, I believe the number is 803424. Naturally, the number for each of the known 26q* positions is 1/3 this number, or 267808. We can further break down these maneuvers into maneuvers that are symmetrically similar. So the total number of symmetrically distinct 26q* maneuvers for these positions is 16738.

I used Tom Rokicki's rubsolv program (version 163) to solve many positions 3q away from one of the known 26q* positions. I wrote some perl code to allow me to take the positions produced by Tom's program, expand them to all symmetrical variants, recanonicalize those maneuvers, and eliminate duplicates.

(With the -s option of Tom's program, supposedly a representative of each symmetry class should be generated, but I think it has a flaw as I was not getting representatives of all maneuvers using that mode of the program. This can be more quickly seen with the simpler "four-spot" position that has the same symmetry. So I had to run the program without using the -s option.)