Six vertical mirrors are arranged to form a regular hexagon with only a small gap (of negligible size) at each vertex. A beam of light is shone through one of these gaps toward the one directly opposite, through which it exits.

However, the beam can be aimed slightly off this direct route to the other side, so that it is reflected off each of the mirrors exactly once before finally exiting the arrangement through that same opening opposite its entrance.

What is the smallest angle off of straight across which will make this possible?