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Snatoms and Possible Combinations

I recently came across Snatoms by Derek Muller (Famous Veritasium guy).

It made me think about this question:

Let's say we have a 'C' shape which can have 4 attachments to it

'O' shape that can have 2 attachments to it

and 'H' shape that can have only 1 attachment

How many possible valid combinations can be made out of 6 'C', 6 'O' shapes and 12 'H' shapes? Is there any way to generalize this? (Don't think about chemistry concepts while solving this problem, and by "valid" combinations I mean to exclude combinations that are not possible for instance CH5 to CH12)

Note by Vikram Pandya
1 year ago

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Oh yes! That starts to delve into graph theory. You are basically asking for the number of graphs (up to isomorphism) where the vertices of degree 4, 2, 1 occur at most 6, 6, 12 times.

There are 2 ways to interpret the question
- We care about the positioning of the graph. E.g. \( O = O, O = O \) is considered different from \( O - O - O - O \) (with the last linked back to the first)
- We only care about the vertices that are used. Calvin Lin Staff · 1 year ago

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@Calvin Lin Can you take a stab at generalization of this number theory problem over here Vikram Pandya · 12 months ago

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@Calvin Lin Hey Calvin, thanks for stopping by. I was travelling so could not reply in time. When I posed the question I had only vertices in my mind :) Vikram Pandya · 1 year ago

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