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:

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


 * 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:


 * 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)