This question is about the popular logic game called Tantrix. I would like to collect combinatorial theorems about it, eg. necessary conditions for making a cycle of one color from a given set of tiles that passes through all the tiles and the union of tiles has no holes in it. On the web I could only find theorems about the complexity of the game which is a completely different question. To show what kind of theorems I want, here are three easy observations.
Theorem Trivial. If there is a red cycle using all the tiles, then red must appear on all the tiles.
Theorem Crossing Parity. If there is a red cycle using all the tiles, then the number of red-blue crossings must be even.
Theorem Winding. Count straight red tiles 0 (can be denoted by I), big turns (which are almost straight, can be denoted by L) as 1 and small turns (when red touches adjacent sides, can be denoted by V) as 2. If there is a red cycle using all the tiles, then it must be possible to assign a sign to each tile such that the sum is 6.
I would be also interested in configurations that satisfy these theorems but it is still impossible to make a cycle, like 3 Vs and a non-zero number of Is.