From mark.longridge@canrem.com Sun Oct 8 00:27:14 1995 Return-Path: Received: from itchy.crso.com (itchy.canrem.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA08707; Sun, 8 Oct 95 00:27:14 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1F8133; Sat, 7 Oct 95 23:56:06 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Picture Cubes From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1250.5834.0C1F8133@canrem.com> Date: Sat, 7 Oct 95 23:55:00 -0500 Organization: CRS Online (Toronto, Ontario) David Singmaster (ZINGMAST@VAX.SBU.AC.UK) writes on Wed Sep 13 11:50:09 1995: ------------------------------------------------ I think the ranking is not quite fair because the puzzles are of very different types. E.g. the 15 puzzle has nearly as many patterns as the 3^3 but no one would claim it was anywhere near as difficult. Indeed the Babylon Tower has 36! = 3.72 x 10^41 basic positions. One can divide by some small value such as 2 or 6 or perhaps more, depending on what one considers the same position. This puts it between 3 and 4 in your list, but it is not a difficult puzzle, except that it is hard to see the gradations of the colors! Indeed, the commercial 7 x 7 'fifteen puzzles' have 49! =6.08 x 10^62 basic patterns - again one has to divide by something, in this case 2. This falls between 2 and 3 in your list, but again it is hardly a difficult puzzle. So I think you are comparing puzzles which are of such different type that the number of patterns is not a fair comparison of their difficulties.I would group them in three (or perhaps 2) types. Rubik Cube, etc. Fifteen Puzzles, etc. in the plane. Cylindrical Puzzles - barrels, etc. ------------------------------------------------ I quite agree. One of my reasons for making that list was to simply rank all the puzzles by number of combinations only, to show the feasibility (or lack of) for a brute force search to find God's Algorithm. The major drawback, as you point out, is that difficulty in solving is not only a function of the number of combinations. Dr. Singmaster continues: ------------------------ Re your Case E. Almost all the picture cubes have all four orientations distinct on the face centres - both those with nine little pictures on each face and those with a big picture spread over all nine facelets. These are actually pretty common. ------------------------------------------------- Case E, that is cases that have only a fraction of the total possible number of combinations for a Rubik's picture cube, are unfortunately well represented in my own cube collection. The following cubes are all in Case E: Rubik's Calendar Cube, Rubik's Cube 4th Dimension, Rubik's World, Blind Man's Cube (from Germany), Royal Wedding Cube (with Charles & Di). Although I don't doubt that, over all, these cases are exceptional. In the case of Rubik's World there are 3 blank centre pieces, and in the Royal Wedding Cube only 2 opposite faces can show all 4 possible orientations. Name Combinations Inventor 8. Picture Cube (3x3x3) (E) 8.8*10^22 Erno Rubik, Dan Hoey 9. Calendar Cube (3x3x3)(F) 4.4*10^22 Marvin Silbermintz 10. Rubik's Cube 4th Dim.(D) 1.1*10^22 Erno Rubik 11 Rubik's World (G) 2.7*10^21 Erno Rubik 12. Royal Wedding Cube 6.9*10^20 Unknown 13. Rubik's Cube (3x3x3) 4.3*10^19 Erno Rubik -> Mark <-