I was thinking about hexagons and triangles and came up with the following problem:

Given an equilateral triangle, derive a formula for the minimum number of non-overlapping regular hexagons (can be of any side length, but must be fully contained within the triangle) needed to fill at least n of the triangle's area, where n is a real number [0, 1).

Intuitively, it seems natural to place the big dark gray hexagon first, making the big triangle into 3 smaller ones, each of side length 1/3 of the the big one. Then repeat this process recursively with the light gray hexagons and so on until you reach n.

I might be smooth brained, but I can't seem to prove that this is the most efficient way. Any ideas, /sci/?