From ronnie@cisco.com Wed Jul 28 19:37:15 1993 Return-Path: Received: from lager.cisco.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00159; Wed, 28 Jul 93 19:37:15 EDT Received: from localhost.cisco.com by lager.cisco.com with SMTP id AA06770 (5.67a/IDA-1.5 for ); Wed, 28 Jul 1993 16:37:08 -0700 Message-Id: <199307282337.AA06770@lager.cisco.com> To: Cube-Lovers@life.ai.mit.edu Subject: Hint wanted for 4x4x4 Date: Wed, 28 Jul 1993 16:37:08 -0700 From: "Ronnie B. Kon" I've been beating my head against the order 4 Rubik's cube for long enough, and I want a hint. (Not a solution--I have a solution book if I wanted to use it). My problem is I cannot flip a pair of adjacent edges (this is equivalent to not being able to exchange a pair of knights-move separated edges). All my other transformations have no side effects, so I can solve the edges first. But I can't see how to just affect two of them. I tend to solve using commutators, but I don't see a way here. The move I use on the top moves the marked pieces clockwise (this pattern . . 0 . . . . . . . . 0 . 0 . . rotates and reflects, of course). There is no way to combine these into a pair exchange (after doing the move, you still have two pieces out of place--nothing changed from the original). I tried to find a move that would exchange three pieces, the third being the correctly placed piece next to one of the incorrectly placed pieces (ie., treat a right edge cubie as if it should be a left edge cubie) but this can easily be shown as impossible: Define the parity of a piece as being left if it is a left edge cubie when the red facelet is up, right if it is a right edge cubie when the red facelet is up. The parity is undefined if there is no red facelet. There are only three moves available that affect an edge cubie--none of them alter the parity. QED So, what am I missing? As I said before, I really just want a hint here. Ronnie