Date: Tue, 31 Oct 95 01:02:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Spotty Megaminx Revisited
```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 <-
```