19 arrows hit the target in the form of a regular hexagon page length of 1 m. Show that at least two arrows are less than 60 cm away.

Question

Problem:
19 arrows hit the target in the form of a regular hexagon page length of 1 m.
Show that at least two arrows are less than 60 cm away.

My attempt: Idea is to use Pigeonhole principle to solve this problem. In this type of problem we need to divide triangle in little triangle, and square in little square. So I don't have idea how to deal with hexagon.

Answer 1

We divide the hexagon into six equilateral trainanles of side length $1\,\mathrm m$. Divide each triangle into three kite-like shapes by cutting along the perpendiculars from the triangle centre to the edges. At least one of the eighteen kites is hit by two arrows, and the diameter of such a kite is $\frac23\cdot\frac{\sqrt 3}2\,\mathrm m\approx 57.7\,\mathrm{ cm}<60\,\mathrm{cm}$.