How do i get the number of vertices of a grid of n hexagons?

Question

What is the equation for finding the number of vertices in a grid on n hexagons

Answer 1

You can only give bounds for this number.

 

Let's assume, the n hexagons must form a polyhex (like a polyomino but with regular hexagons).

 

Maximum:

 

The first hexagon contributes 6 vertices and each new hexagon after that contributes at most 4 vertices.

 

MAX = 6 + 4(n-1) = 4n+2

 

Minimum:

 

You need to maximize the number of hexagons which are completely surrounded. See sequence A121149 in the Online

Encyclopedia of Integer sequences for the minimum. There is unlikely to be a nice formula for the minimum.

 

As we add hexagons, we add more and more hexagons, we find that the vast majority of the new hexagons will contribute 2 new vertices each. Specifically, the new hexagons that contribute more than two become exceedingly rare and there are never new ones contributing only 1 new vertex.

 

Thus:

 

lim_n→∞ MAX(n) / MIN(n) = 2