Has the diameter of the Rubik's cube puzzle been established yet?

i don't believe so. before approaching this question, one needs to decide how to measure distance. there are two common approaches, the "face turn" metric and the "quarter turn" metric. in the face turn metric, you can turn any outer face as far as you want, and it counts as 1 turn. in the quarter turn metric, each 90 degree rotation of an outer face counts as 1 turn. the quarter turn metric has the nice property that the cayley graph is bipartite. for this (and other reasons) i think it is the most natural metric to consider. i did some work on this problem way back when. in 1995, i showed that 42 quarter turns and 29 face turns are upper bounds for the diameter in the respective metrics. i also showed that the position "superflip", in which all 12 edges are flipped in place, is exactly 20 face turns from solved, and is either 22 or 24 quarter turns from start. each of these improved the known lower bounds for the diameter, which at the time were due to counting (i.e. existential) arguments. jerry bryan later showed that superflip is indeed 24 quarter turns from solved. in 1998, i discovered a related position that is exactly 26 quarter turns from solved. this position is "superflip composed with four spot"; i.e. first create the position superflip, then perform a maneuver for the pattern "four spot", which has two solved faces and four faces in which the central square is the wrong color. exercise for munsters: show that this position is locally maximal in the quarter turn metric, that is to say, any quarter turn moves you closer to the solved position. i have heard that some improvements have been found for the upper bounds, but i'm unaware of any improvements to the lower bounds. i haven't followed along since the demise of the cube-lovers mailing list, and those involved in the improvements have not gone out of their way to keep me up to date. some information is at: http://cubezzz.homelinux.org/drupal/ optimal cube solvers are available, my own is at http://www.math.ucf.edu/~reid/Rubik/optimal_solver.html and has been available since 1997. it has only a primitive text interface, but should compile and run on any ansi c system. i have another solver that uses much larger pattern databases (~883Mb) and i was recently requested to make that one available too ... but that will be on an "as time permits" basis. if you like gui interfaces and use micros**t, you should try herbert kociemba's optimal solver http://www.kociemba.homepage.t-online.de/cube.htm which has some other really cool features. mike