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I read the articles in the archives about Kociemba's algorithm about

a year ago, without (I confess) fully understanding them. In particular,

I do not fully understand what differentiates Kociemba's algorithm from

Thistlethwaite's algorithm, other than it uses a different arrangement

of nested subgroups.

The basis is similar (although Kociemba's algorithm uses searching to

get solutions while Thistlethwaite's uses tables and directly arrives

at solutions). The main difference is that once a solution is found

Thistlethwaite's algorithm stops. Kociemba's algorithm continues finding

newer solutions (even longer than the original solution) to phase 1 and

trying to fit them with a solution for phase 2 such that the total solution

is shorter. This proves to be very effective. Of course this is easier to

do with a 2 phase algorithm than with a 4 phase algorithm.

But in the meantime, I wonder if you could verify that Kociemba's

algorithm does not guarantee to find a minimal process? In particular,

is it the case that 26q is a minimal superflip, or is it only an

upper bound?

Given time Kociemba's algorithm will find a minimal solution. I confess

that my implementations does not if the configuration can be solved

through phase 2 only, but the cube can be rotated to avoid that. The

reason is that ultimately Kociemba's algorithm will find longer part

solutions of phase 1 and ultimately stumble on a complete solution in

phase 1 which will be minimal because of the breadth first search.

But it can take long. Getting a minimal solution if the length is 16

or less appears to be doable. If the length is 19 or more it takes an

awfully long time. What I have found until now is:

1. There are no configurations known that require 21 turns or more,

and I checked an awfully large number.

2. There are known configurations that require 18 turns.

The middle part is a grey area.