The Math Behind The Game:

Inspired by a Discrete Mathematics homework problem: the concept of how many possible ways can edges exist between dots (in this case 4) arranged in a circle (or a square). Organized in an efficient manner, we can see many patterns. 

1) I will tell you that there are 63 such possibilities, 21 unique to rotation, and 18  unique to both rotation and reflection (excluding the possibility of zero edges). With four dots, we know the maximum number of edges you can have is 6 (a square with an X). After finding all combinations, and organizing them by number of edges, we can see this pattern: for zero edges, we have 1 possibility. for one edge, we have 6. for two, we have 15, three gives 20, four gives 15 again, five gives 6 and six gives 1. And these numbers are exactly the coefficients of the polynomial (x + 1)^6.

2) We can find another pattern. Pick an arbitrary combination that does not contain 3 edges, let's say n edges. The 6-n edges missing in this combination will correspond to another combination with 6-n edges. For example, let's pick the square, it has four edges. What's missing? The X, which has 6-4=2 edges. Now what if we choose a combination with three edges. We know there are 5 combinations with three edges unique to rotation and reflection. The U shape needs the X with a fedora, the arrow needs an arrow head pointing in the opposite direction, but what about the N? What's most interesting about the N is that it is the only combination that is missing itself. Not its reflection, but itself (rotated on its side). And I find it funny that, in my opinion, the N is the hardest to play with in the game. 


Bugs:

These rules may not be followed all the time.

1) Physics - The only pieces that should be allowed to go through each other are slants of opposite orientation. 

2) Tile Clearance - Tiles need all 6 pieces to clear. 

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