From cosell@bbn.com Tue Jan 7 15:48:19 1992 Return-Path: Received: from WILMA.BBN.COM by life.ai.mit.edu (4.1/AI-4.10) id AA06487; Tue, 7 Jan 92 15:48:19 EST Message-Id: <9201072048.AA06487@life.ai.mit.edu> Date: Tue, 7 Jan 92 15:02:18 EST From: Bernie Cosell To: cube-lovers@life.ai.mit.edu Subject: Re: Hungarian Rings solution? In rsponse to my request for info about the Hungarian Rings, Dik.Winter@cwi.nl writes: } You don't nood commutators for it, cycles are sufficient (because there Dik.Winter@cwi.nl writes: } You don't nood commutators for it, cycles are sufficient (because there } are so many similar colored beads).... My apologies --- I meant to say "cycles" when I said I had found lots of them... And I hate to seem dense, but but I'm still stuck... } ... If I remember right one useful move } is: turn right ring clockwise two beads, turn left clockwise two beads, } turn right anti-clockwise two beads, turn left anti-clockwise two beads. } Using them properly will solve the rings. The 'properly' is the part I'm finding hard. There seem to be LOTS of cycles, but even with that big choice I can't see, quite, how to solve the thing. As far as I can tell, basically ANY set of ring-turns that has a total movement of zero seems to define a pretty small cycle. For example, the sequence LnA RnA LnC RnC, for n not a multiple of 5[*], does a three-bead cycle: if you look at the upper intersection: A C Intersection ---> C ======> B B A Where 'A' and 'B' are each n beads away from the intersection [and by changing theorder of L/R you reverse the cycle, and by interchanging A and C you move the cycle to the other side of the intersection. BUT: the problem is that this isn't really a 3-cycle, but rahter _two_ 3-cycles: you also make a central-symmetric move of the beads at the bottom intersection. [*] since five is the distance between the intersections, if the rotate is a muiltiple of 5 the intersections interact, things get a little different: it makes a *two* cycle! In the diagram above [with A five away from C], the move just _swaps_ A & C [and the A' and C' at the lower intersection, too, of course]. Given that my rings are totally non-symmetrically messed up, I can't figure out a plan for making forward progress. I can do lots of diffent cycles, but I can't manage to get the rings set up so that the cycle at both intersections is useful: if I try to fix something at the top intersection I invariably mess up something at the bottom one. Thanks again for you patience with my rantings. I feel like I'm overlooking something simple [since this wasn't supposed to be all that hard a puzzle], but I don't see what it is ... /Bernie\