Topic, The topic where you can only post the most random things you can think of that doesn't make sense...

I do enjoy doing those kinds of puzzles. That one specifically relates to Centered Polygonal Numbers (in this case Centered Hexagonal Numbers), and I think learning how to work out the U(n) in a recurrence relation if U(0)=a, U(n)=U(n-1)+bn is an interesting exercise. There are non-centered ones (triangular and square numbers are very commonly discussed types of numbers, and triangular numbers can be used to solve for U(n) as above), and also numbers that are based on 3D shapes, obviously cube numbers but other platonic solids as well. At some point studying them gets too abstract (and too pointless) for a lot of people.

H(1)=1, H(n+1)=H(n)+6n
H(n)=6n(n-1)/2+1=3n^2-3n+1

H(0)=1 (this makes no intuitive sense, with zero length there should be zero area)
H(-1)=7 (???)
H(i)=-2-3i

These numbers don’t make sense, and I’m sure there’s other numbers you can match up to the original puzzle, but that is the method I found.

I think the areas you want are 7 and 19, so 2 and 14 more than the 5 you start with should be enough to make these hexagons.