From hoey@aic.nrl.navy.mil Mon Aug 3 11:10:35 1992 Received: from Sun0.AIC.NRL.Navy.Mil by life.ai.mit.edu (4.1/AI-4.10) id AA00431; Mon, 3 Aug 92 11:10:35 EDT Received: from sun13.aic.nrl.navy.mil by Sun0.AIC.NRL.Navy.Mil (4.1/SMI-4.0) id AA22147; Mon, 3 Aug 92 11:10:30 EDT Return-Path: Received: by sun13.aic.nrl.navy.mil; Mon, 3 Aug 92 11:10:29 EDT Date: Mon, 3 Aug 92 11:10:29 EDT From: hoey@aic.nrl.navy.mil Message-Id: <9208031510.AA03178@sun13.aic.nrl.navy.mil> O: Mark Longridge Cc: Cube-lovers@life.ai.mit.edu Subject: Re: cube theory Dear Mark, I sent you email on 8 June. Did you receive it? Are you interested in acquiring a Tartan cube? Did you ever get Minh Thai's book on Rubik's Revenge? With respect to the 3x3x3 squares group, > How do you keep track of all the patterns without repeating yourself? there are several ways. The most general is to use the Sims table (aka the FHL table) for the subgroup, which gives a mixed-base enumeration of the positions. See my message of 1 February 1981. > A local maxima is a state where any possible move will bring you > closer to a solution. That's ``local maximum''. ``Maxima'' is the plural of ``maximum''. > Note that all possible patterns at maximum depth are local > maxima, .... We call such positions ``global maxima'' because they are at the overall maximum depth. The statement is then that every global maximum is a local maximum. > To date, no work has been done to determine the depth of the > dodecahedron (megaminx).... Well, I've done some looking at it. Since my initial remarks on 23 September 1982, I've figured out a way to generate a recurrence for it, but it seems I haven't put it down anywhere. Are you interested? (Do I have to tell anyone to answer that question only to Hoey@AIC.NRL.Navy.Mil, not the list?) > What pattern is an example of local maxima? e.g. 3x3x3 at depth 3 > -> 12-flip, 12-flip 8-twist Jim Saxe and I listed the 25 symmetric local maxima in our message on Symmetry and Local Maxima, dated 14 December 1980. We verified Jim's conjecture that the four-spot is a local maximum, but not on the grounds of symmetry, and reported that on 22 March 1981. Do you have access to the cube-lovers archives? > Moves Deep arrangements (q+h) arrangements (q only) * > 0 1 1 > 1 18 12 > 2 243 114 > 3 3,240 1,068 > 4 > 48,600 10,011 This is in the archives, too 5 93,840 (22 March 1981) 6 878,880 (14 August 1981) 7 8,221,632 (7 December 1981) David C. Plummer and I had hoped to use his program (which counted the 7 QT positions) to extend this to 8 QT, but we got busy. I still have hopes.... > At 2 moves deep it is redundant to turn the same side again.... > However, with opposite turns order is not significant, e.g. T,D = D,T.... This approach appeared on 9 January 1981. It showed how to follow the argument to 25 QT, and to get what are still the best known lower bounds for the ordinary cube and for the supergroup. > Also what is an example of a local maxima close to the surface, e.g. > 4 moves. I believe Jim Saxe and Dan Hoey have done some work in this > regard. It's known there are no local maxima at 7 QT or less. The shortest known local maxima are Pons Asinorum and the four-spot, both at 12 QT. I know of no results between 8 and 11 QT. Dan Hoey Hoey@AIC.NRL.Navy.Mil