From mark.longridge@canrem.com Tue Oct 31 01:27:02 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 AA09622; Tue, 31 Oct 95 01:27:02 EST Received: by canrem.com (PCB-UUCP 1.1f) id 1FBDC8; Tue, 31 Oct 95 01:11:34 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Spotty Megaminx Revisited From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1257.5834.0C1FBDC8@canrem.com> Date: Tue, 31 Oct 95 01:02:00 -0500 Organization: CRS Online (Toronto, Ontario) Notes on the Spot Patterns on the Megaminx ------------------------------------------ Number the faces of the megaminx 1 through 12. Here are all the possible permutations of the 12 centres: dod := Group( (2,3,4,5,6) (7,8,9,10,11), (1,4,10,9,2)(5,11,12,8,6) );; Size (dod) = 60; NumberConjugacyClasses (dod) = 5; Elements (dod); [ (), 0 spot ( 2, 3, 4, 5, 6)( 7, 8, 9,10,11), 2 5-cycles = 10 ( 2, 4, 6, 3, 5)( 7, 9,11, 8,10), 2 5-cycles = 10 ( 2, 5, 3, 6, 4)( 7,10, 8,11, 9), 2 5-cycles = 10 ( 2, 6, 5, 4, 3)( 7,11,10, 9, 8), 2 5-cycles = 10 ( 1, 2)( 3, 6)( 4, 8)( 5, 9)( 7,10)(11,12), 6 2-cycles = 12 ( 1, 2, 3)( 4, 6, 9)( 5, 8,10)( 7,12,11), 4 3-cycles = 12 ( 1, 2, 6)( 3, 8, 5)( 4, 9, 7)(10,12,11), 4 3-cycles = 12 ( 1, 2, 8, 7, 5)( 3, 9,12,11, 4), 2 5-cycles = 10 ( 1, 2, 9,10, 4)( 5, 6, 8,12,11), 2 5-cycles = 10 ( 1, 3, 2)( 4, 9, 6)( 5,10, 8)( 7,11,12), 4 3-cycles = 12 ( 1, 3, 9, 8, 6)( 4,10,12, 7, 5), 2 5-cycles = 10 ( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,12)( 8,11), 6 2-cycles = 12 ( 1, 3,10,11, 5)( 2, 9,12, 7, 6), 2 5-cycles = 10 ( 1, 3, 4)( 2,10, 5)( 6, 9,11)( 7, 8,12), 4 3-cycles = 12 ( 1, 4,10, 9, 2)( 5,11,12, 8, 6), 2 5-cycles = 10 ( 1, 4,11, 7, 6)( 2, 3,10,12, 8), 2 5-cycles = 10 ( 1, 4, 3)( 2, 5,10)( 6,11, 9)( 7,12, 8), 4 3-cycles = 12 ( 1, 4, 5)( 2,10, 7)( 3,11, 6)( 8, 9,12), 4 3-cycles = 12 ( 1, 4)( 2,11)( 3, 5)( 6,10)( 7, 9)( 8,12), 6 2-cycles = 12 ( 1, 5, 7, 8, 2)( 3, 4,11,12, 9), 2 5-cycles = 10 ( 1, 5, 6)( 2, 4, 7)( 3,11, 8)( 9,10,12), 4 3-cycles = 12 ( 1, 5,11,10, 3)( 2, 6, 7,12, 9), 2 5-cycles = 10 ( 1, 5, 4)( 2, 7,10)( 3, 6,11)( 8,12, 9), 4 3-cycles = 12 ( 1, 5)( 2,11)( 3, 7)( 4, 6)( 8,10)( 9,12), 6 2-cycles = 12 ( 1, 6, 2)( 3, 5, 8)( 4, 7, 9)(10,11,12), 4 3-cycles = 12 ( 1, 6, 8, 9, 3)( 4, 5, 7,12,10), 2 5-cycles = 10 ( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,11)(10,12), 6 2-cycles = 12 ( 1, 6, 5)( 2, 7, 4)( 3, 8,11)( 9,12,10), 4 3-cycles = 12 ( 1, 6, 7,11, 4)( 2, 8,12,10, 3), 2 5-cycles = 10 ( 1, 7, 2, 5, 8)( 3,11, 9, 4,12), 2 5-cycles = 10 ( 1, 7, 9)( 2, 6, 8)( 3, 5,12)( 4,11,10), 4 3-cycles = 12 ( 1, 7,10)( 2, 8, 9)( 3, 6,12)( 4, 5,11), 4 3-cycles = 12 ( 1, 7)( 2,11)( 3,12)( 4, 8)( 5, 6)( 9,10), 6 2-cycles = 12 ( 1, 7, 4, 6,11)( 2,12, 3, 8,10), 2 5-cycles = 10 ( 1, 8, 3, 6, 9)( 4, 7,10, 5,12), 2 5-cycles = 10 ( 1, 8)( 2, 6)( 3, 7)( 4,12)( 5, 9)(10,11), 6 2-cycles = 12 ( 1, 8, 5, 2, 7)( 3,12, 4, 9,11), 2 5-cycles = 10 ( 1, 8,10)( 2, 9, 3)( 4, 6,12)( 5, 7,11), 4 3-cycles = 12 ( 1, 8,11)( 2,12, 4)( 3, 9,10)( 5, 6, 7), 4 3-cycles = 12 ( 1, 9, 6, 3, 8)( 4,12, 5,10, 7), 2 5-cycles = 10 ( 1, 9)( 2, 3)( 4, 8)( 5,12)( 6,10)( 7,11), 6 2-cycles = 12 ( 1, 9, 7)( 2, 8, 6)( 3,12, 5)( 4,10,11), 4 3-cycles = 12 ( 1, 9, 4, 2,10)( 5, 8,11, 6,12), 2 5-cycles = 10 ( 1, 9,11)( 2,12, 5)( 3,10, 4)( 6, 8, 7), 4 3-cycles = 12 ( 1,10, 8)( 2, 3, 9)( 4,12, 6)( 5,11, 7), 4 3-cycles = 12 ( 1,10, 2, 4, 9)( 5,12, 6,11, 8), 2 5-cycles = 10 ( 1,10, 7)( 2, 9, 8)( 3,12, 6)( 4,11, 5), 4 3-cycles = 12 ( 1,10)( 2,11)( 3, 4)( 5, 9)( 6,12)( 7, 8), 6 2-cycles = 12 ( 1,10, 5, 3,11)( 2,12, 6, 9, 7), 2 5-cycles = 10 ( 1,11, 8)( 2, 4,12)( 3,10, 9)( 5, 7, 6), 4 3-cycles = 12 ( 1,11, 9)( 2, 5,12)( 3, 4,10)( 6, 7, 8), 4 3-cycles = 12 ( 1,11, 3, 5,10)( 2, 7, 9, 6,12), 2 5-cycles = 10 ( 1,11, 6, 4, 7)( 2,10, 8, 3,12), 2 5-cycles = 10 ( 1,11)( 2,12)( 3, 7)( 4, 5)( 6,10)( 8, 9), 6 2-cycles = 12 ( 1,12)( 2, 7)( 3,11)( 4,10)( 5, 9)( 6, 8), 6 2-cycles = 12 ( 1,12)( 2, 8)( 3, 7)( 4,11)( 5,10)( 6, 9), 6 2-cycles = 12 ( 1,12)( 2, 9)( 3, 8)( 4, 7)( 5,11)( 6,10), 6 2-cycles = 12 ( 1,12)( 2,10)( 3, 9)( 4, 8)( 5, 7)( 6,11), 6 2-cycles = 12 ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7) 6 2-cycles = 12 Number Pattern ------ ------- 1 0 spots 24 2 five-cycles (10 spot) 15 6 two-cycles (12 spot) 20 4 three-cycles (12 spot) -- 60 orientations of the dodecahedron, 24 ten-spots, 35 twelve-spots >> I suspect various 12-spots are possible. I have no idea how to >> easily permute centre pieces on the megaminx. > > Indeed. Every rotation of the center skeleton is possible (if you > consider the remainder fixed...). So there are 12 centers that can > come out at top; for each center at top you have 5 possible positions > of the remainder leading to 60 configurations. Of these 24 are > 10-spots, 1 is the solved puzzle, so the remainder (35) is 12-spots. > dik Well, I was confused how there could be 35 twelve-spots (at first), but I am happy to confirm Dik's memory. -> Mark <-