Talk:Serpentiles
Latest comment: 2 years ago by Mliu92 in topic Generalizing to 2n-sided regular polygons
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Generalizing to 2n-sided regular polygons
editSome thoughts:
For any regular polygon with an even number of sides 2n, how many possible combinations exist for n paths connecting paired sides? Assume 2n = M for convenience.
Observations from the 4-, 6-, and 8-sided regular polygons:
- There is always a case linking all adjacent sides (00...0M)
- There is always a case linking all opposite sides (M0...00)
4-sided (squares)
editFor the case where 2n = 4, n = 2 paths link paired sides. There are two possible combinations:
-
02 / 12-34
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20 / 13-24
6-sided (hexagons)
editFor the case where 2n = 6, n = 3 paths link paired sides. There are five possible combinations:
-
003 / 12-34-56
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021 / 15-26-34
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102 / 16-25-34
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120 / 13-25-46
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300 / 14-25-36
- 3 cases have at least one link between opposite sides, including the all-opposite (300) case.
8-sided (octagons)
editFor the case where 2n = 8, n = 4 paths link paired sides. There are eighteen possible combinations:
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0004
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0022
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0040
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0103
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0121(a)
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0121(b)
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0202
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0220
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0301
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0400
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1012
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1111(a)
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1111(b)
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1210
-
2002
-
2020
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2200
-
4000
- 8 cases have at least one link between opposite sides, including the all-opposite (4000) case.
- 4 cases have at least two links between opposite sides. Discounting the all-opposite 4000 case, when there are two pairs of opposite sides linked, that leaves four sides to be linked. The unlinked sides have either adjacency 0 or 1. There are two ways to link the adjacency 0 sides (2002 or 2200), but only one way to link the adjacency 1 sides (2020).
Cheers, Mliu92 (talk) 17:17, 31 May 2022 (UTC)