Talk:Serpentiles

Generalizing to 2n-sided regular polygons

Some 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)

For the case where 2n = 4, n = 2 paths link paired sides. There are two possible combinations:

• 
  02 / 12-34
• 
  20 / 13-24

6-sided (hexagons)

For the case where 2n = 6, n = 3 paths link paired sides. There are five possible combinations:

• 
  003 / 12-34-56
• 
  021 / 15-26-34
• 
  102 / 16-25-34
• 
  120 / 13-25-46
• 
  300 / 14-25-36

• 3 cases have at least one link between opposite sides, including the all-opposite (300) case.

8-sided (octagons)

For the case where 2n = 8, n = 4 paths link paired sides. There are eighteen possible combinations:

• 
  0004
• 
  0022
• 
  0040
• 
  0103
• 
  0121(a)
• 
  0121(b)
• 
  0202
• 
  0220
• 
  0301
• 
  0400
• 
  1012
• 
  1111(a)
• 
  1111(b)
• 
  1210
• 
  2002
• 
  2020
• 
  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)