Mail-from: SU-NET host SU-SHASTA rcvd at 16-Aug-82 1204-PDT Date: Monday, 16 Aug 1982 12:04-PDT To: cube-lovers at Mit-mc Subject: Cube Mechanics From: Tom Davis I finally found a 4x4x4 cube a couple of days ago, and have a couple of interesting observations. Forgive me if I repeat anything said so far, but I have been ignoring everything on this list having to do with the 4^3 cube for fear of any sort of spoilers. Using my 3^3 knowledge, I found it fairly easy to get it almost solved. Half the time, however, I got it to the state where everything was solved except that two adjacent edge cubies were flipped. I finally convinced myself by means of a somewhat involved (and probably fallacious) "proof" that I would have to exchange them before I could solve the cube. My first observation is simply a trivial proof of that fact that I discovered immediately after I took the cube apart for the first time to see what was inside -- it is mechanically impossible to put the cube back together with the cubies flipped (but not exchanged). Some similar parity-type arguments can be made about possible configurations of the center cubies. What is interesting is that this presents a new method of proving things about configurations -- if one can dream up a mechanical model of a cube with different guts, it may be obvious that some sorts of things are impossible. The cube simply has to behave the same way externally. I wonder if there are nice ways to look at the various parity-trinity features of the three-cube by looking at it using a different model of the internal mechanics. My second observation is that although a 5^3 and a 6^3 may someday appear on the market, the 7^3 will be pretty tricky to build. When one of the faces of a 7^3 is turned 45 degrees, the corner will lie completely outside the original cube. Any mechanical linkage will be complicated indeed. Maybe a cube could be built with little microprocessors inside each cubie controlling little arms and hooks to grab adjacent cubie faces ... -- Tom Davis