Tantrix from combinatorial viewpoint This question is about the popular logic game called Tantrix.1 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.
1 See also the Wikipedia article on Tantrix.
 A: I am trying to figure out the combination of pieces needs to form a loop so that solutions are easier to conceptualize before trying to play with the pieces.
The tight turns (V) are 120 degrees and the less tight turns (L) are 60 degrees and a loop must contain a total of 360 degrees so only specific
i.e. 3V facing inwards (plus any even number of Vs or Ls used in opposite orientations)
2V plus 2L facing inwards (plus any even number of Vs or Ls used in opposite orientations)
1V plus 4L facing inwards (plus any even number of Vs or Ls used in opposite orientations)
There must be similar logic for the lengths in each direction that would indicate the need orientation of the straight pieces and ratio of Vs and Ls on all sides
Any thoughts?
Moksha
A: You might wish to look at some papers by Paul Zinn-Justin.  For example "Littlewood--Richardson coefficients and integrable tilings" defines a model of random tilings which count the Littlewood-Richardson coefficients.  As early as 2001, puzzles were being used to compute the cohomology of complex Grassmanians.  See "The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone" by Allen Knutson and Terry Tao.
Another might try looking in relation to the Temperley Lieb-Algebra or other Planar Algebras.
