OIM 2007 Problem 6

Let $\mathcal{F}$ be a family of hexagons $H$ satisfying the following properties:

i) $H$ has parallel opposite sides.

ii) Any 3 vertices of $H$ can be covered with a strip of width 1.

Determine the least $\ell\in\mathbb{R}$ such that every hexagon belonging to $\mathcal{F}$ can be covered with a strip of width $\ell$ .

Note: A strip is the area bounded by two parallel lines separated by a distance $\ell$ . The lines belong to the strip, too.