This question has been raised a bunch and I often make some statements with estimations, simply resolving to say, a lot. But this is a fun math problem so I thought I would share and start here. I’m not going to solve the problem in this post, but wanted to get the wheels turning.
First, I thought the right place to start would be the possible physical arrangements of 6 Blinks. Puzzle101 is designed to work with 6 Blinks, no more, no fewer (and if you are curious, yes there is a prototype of one that does more… so ping us with interest ). I looked up Pentaminoes, which are the arrangements of 5 squares, a type of polyamino like Tetris, which uses arrangements of 4 squares or tetraminos, and eventually found my way to polyhexes.
I figured someone had already done the math to establish how many arrangements of 6 hexagons there are and sure enough, spoiler alert, the answer is ### 82 ###.
Knowing that, then we need to compute the different possible arrangements of 3 colors on matching faces for each of the possible combinations. The fewest possible connections for all 6 Blinks to be touching would be 5 shared edges and the most connections would be 9 shared edges. like the following two highlighted arrangements:
Using one of 3 options for each of 5 to 9 placements is a straightforward combinatorial problem, which I believe looks like 3⁵ = 243 possibilities for coloring or 3⁹ = 19,683 possibilities for coloring.
Without summing up the 82 exact arrangements and calculating their possible color allocations (and perhaps removing colorings that are isomers) we can assume that the final answer is at least 19,926 and less than 1,614,006.
I’d be curious to see your take on figuring out how many puzzles there are, but there sure are a lot. I’m curious to hear what some of your fastest solves are. I think I average ~15 seconds per puzzle