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Here’s a numerical solution in Python (based on my less pretty Perl code that I wrote at the time). It generates closer and closer approximations to the answer and runs in 45ms.

Solution:The angle is approximately 9 degrees.A geometrical way to demonstrate the solution (and to get an exact answer) is to reflect the hexagons rather than the beam of light.

The beam passes through (is reflected at) each of the mirrors (as can be seen by the numbering in the diagram):

The exact answer is then:

Which is (approximately) 8.948275564627°.