From cube-lovers-errors@curry.epilogue.com Sat Nov 16 21:51:20 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id VAA00394; Sat, 16 Nov 1996 21:51:19 -0500 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Sat, 16 Nov 1996 07:24:20 -0500 (EST) From: Nicholas Bodley To: Cube-Lovers@ai.mit.edu Subject: Non-cubical Rubik cousins; physical realizability In-Reply-To: <13Nov1996.162951.Alan@LCS.MIT.EDU> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII (My subject line is a spur-of-the-moment phrase; not deeply considered.) I just got to wondering whether some people have considered theoretical larger analogs of the Magic Domino (btw, would somebody please manufacture some Magic Dominoes? Binary Arts?). To get back on topic, these would be Cube-like puzzles with such "cubie counts" ("Dimensions") as 3X3X4, 3X4X4, etc. Whether these are trivial, I haven't yet thought out; making real, physical ones might not be simple. If this topic is covered in the archives, I apologize; in such a case, could someone recommend non-obvious keywords or names? Is there an agreed-upon concise way of defining the "size/count/dimensions" of a Cube; i.e., a 2X2X2 is a Pocket Cube, a 4... is Rubik's Revenge, etc.? How about "order-3" for a regular Rubik's, or simply (given proper context) "n", so that "2" signifies Pocket, "4" Revenge, etc.? Perhaps it's just a personal reaction, but I find it cumbersome to type "5X5X5" more than a few times, for instance. Thinking about this brings up another topic, and probably a difficult one to completely characterize. Given any arbitrary puzzle composed of cubies, is it always possible to create a mechanism to realize that specific puzzle physically? As far as I know (and here I stick my neck waaaaay out!), there is no theory of mechanisms in the general case that would, for instance, say whether an order-2 is realizable (as we know, it can be made, and has been); the Magic Domino is more of a challenge, imho, in that it isn't as easy to say whether such a structure can be made. Some matters affecting realizability are relatively easy to anticipate, such as the matter of holding the corner cubies in place in a "7" (with all cubies of equal size) when one plane is rotated with respect to the other six. Other matters are a question of what's reasonable to design mechanically; while theoretically possible, some structures might not be at all practical, because of such problems as cumulative friction, lack of rigidity, and dimensional tolerances. Such real-world considerations (unfortunately!) muddy the waters until a really good mind comes along to settle the mud. A preliminary guess at an answer to the question is that probably all "low-order" collections of cubies are realizable, but we are far from having a theory of mechanisms that tells us how to design the innards. I maintain that the mechanism of the ordinary Rubik's Cube is the most ingenious simple one ever invented; I have studied mechanisms to a fair degree. (A good competitor is the programmable pushbutton combination lock that has five buttons in a row. This is mechanical, digital, programmable, combinatorial, and sequential.) Hope and trust this hasn't been a waste of bitspace! |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* When the year 2000 begins, we'll celebrate |* Amateur musician *|* the 2000th anniversary of the year 0. --------------------------------------------------------------------------