Why am I staying up so late for this? I find patterns irresistible – that has to be it.

In my preliminary research on symmetry, I’m finding the math will be much more complex when limiting tiles that look completely different to a human.

For example, let’s say 4 paths will use this algorithm (node-x + 4). The 4 paths are identified as nodes 1-5, nodes 2-6, nodes 3-7, nodes 4-8. Then the two remaining paths could have their own repeatable pattern of (node-x + 1) or they will not follow any repeatable algorithm. Those remaining two paths can only be situated in two ways. For the example, I’ll say the paths are nodes 9-12 and 10-11. If you turn the tile and write down the new resulting paths, mathematically there will be 4 unique path sets. But the tile “looks” the same to a human, so there is only 1 tile that will be used in the game. Even if you look at the mirror image of this tile, it’s the same, again leaving only one tile.

Now for an example where the mirror image is different. The tile is not symmetrical, yet some paths will use the same algorithm. For 4 of the paths, I used two different repeatable algorithms (node-x + 1 and node-x + 5) and the 2 remaining paths had non-repeating algorithms. The first set of these paths are:

Nodes 1-2

Nodes 3-4

Nodes 5-10

Nodes 6-11

Nodes 7-9

Nodes 8-12

Mathematically there are 6 sets of unique paths in that example. But the tile looks the same any way it’s turned. Because the tile is not symmetrical, the mirror image looks slightly different – this results in 6 more unique path sets. So there will only be two tiles used in the game, not 12 as math would suggest.

Math calculations for the true number of tiles in the game can be found based on these 3 dynamics:

-The number of paths that use the same algorithm

-The number of paths that use different algorithms than any other path

-The decision about symmetry (the mirror image thing)

However, I have not figured out the math. I have only played with little hexagonal cutouts and noted my human-view findings.

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