# In search of: 21f*s and 20f*s; a four month odyssey.

At this point, I have found no 21f* positions, but with Herbert Kociemba and Silviu Radu, have found 11,313 (mod M+inv) 20f* positions. This set represents 16,510 mod M positions, and 428,982 overall cube positions. The majority of the positions were found by Silviu using a spectacular coset solver that he will write about soon.

In addition to running my coset solver in a mode to find 20f*'s, I also have been running it in a faster, two-phase mode to prove that there are no 21f*'s in that coset. So far I have finished running over 1,000 cosets in this two-phase manner. These cosets are of size 19 billion positions, and depending on the symmetry of the generating position can represent up to 1.9 trillion total positions each. In total, the cosets I have run represent 7.6e14 positions out of 4e19, or about one cube in 53,000---and there are no 21f*'s in all these positions.

All the technology I have been using has been previously presented on this list. Silviu has some new technology he will be sharing with us soon. In addition, Kociemba has implemented (and perhaps improved) some of the technology himself.

The total set of found 20f*'s are available at http://tomas.rokicki.com/all20.txt.

I plan to continue running both efforts (the 20f* search and the 21f* search) for a little while until something else comes up to occupy the machines. At some point I may turn my attention to the quarter-turn metric to see if there are any other 26q*'s, or perhaps a 27q*.