I am sure a number of people in this group would be interested in knowing about some results that have been posted on the Cube Lover's forum (http://cubezzz.homelinux.org/drupal/) regarding the quarter-turn metric (QTM).
Someone on that forum, identifying himself as Silviu, posted that he had found that the corners can always be solved in 22 quarter turns or less without any net change in the location or orientation of the edges. In January, 2004, Tomas Rokicki had posted to this Yahoo group that the edges can always be solved in 18 quarter turns. Together, this shows that Rubik's cube can always be solved in two phases in no more than 18 + 22 = 40 quarter turns. I have personally verified Silviu's result using a C++ program. I have determined the positions of distance 17 and 18 in Rokicki's edges-only analysis. I have considered all the positions (unique with respect to the symmetries of the cube) where those edge configurations are combined with all possible corner configurations with the same permutation parity as the edge configuration. I have found that these positions are solveable using no more than 38 quarter turns. Since all the remaining positions of the cube can be solved with the above-mentioned two-phase method with no more than 16 + 22 = 38 quarter turns, this shows that all positions of Rubik's cube can be solved in no more than 38 quarter turns. The upper bound for the QTM has been considered to be 42 for the last several years, to my understanding. Verification of these new results will thus establish a new upper bound of 38 for the QTM. It may be possible to further consider positions of distance 16 in the edges-only analysis and show that 37 is also an upper bound. - Bruce [Non-text portions of this message have been removed] ------------------------ Yahoo! Groups Sponsor --------------------~--> Get fast access to your favorite Yahoo! Groups. Make Yahoo! your home page http://us.click.yahoo.com/dpRU5A/wUILAA/yQLSAA/MXMplB/TM --------------------------------------------------------------------~-> Yahoo! Groups Links <*> To visit your group on the web, go to: http://groups.yahoo.com/group/speedsolvingrubikscube/ <*> To unsubscribe from this group, send an email to: [EMAIL PROTECTED] <*> Your use of Yahoo! Groups is subject to: http://docs.yahoo.com/info/terms/
