Thank you and yes I understood it enough to solve the problem now. Super cool!

Thank you for the diagram of the 12th hexagon. As soon as you posted it the solution you were looking for presented itself to me, and ultimately led to the slightest change in my formula. Amazing that the formulas could be so close and yet so far from each other! I really like this and it's the second time I've managed to solve a 6kyu problem!

I'm working on this problem and I can't figure out how there are 12 hexagons in n=6 triangle. I count 10 if we only include length 1 hexagons, and 11 if we include the larger (length = 2) hexagon. I have a formula for finding hexagons with length 1 and also have a formula which I believe solves for all hexagon sizes. Am I missing something obvious? (I drew the triangle and counted the hexagons just to be sure).

I'm glad I didn't know this existed.