**From New Scientist #2695, 14th February 2009** [link]

Joe has been showing Penny a few optical experiments. In one experiment he placed six mirrors vertically to form a regular hexagon with small gaps between the mirrors. Through one gap he shone a laser beam so it emerged straight from the gap diametrically opposite.

Penny then had to work out the smallest angle through which the beam must be rotated so that it emerged from the same gap as before, after being reflected just once by all six mirrors.

What was that small angle (rounded to the nearest degree)?

[enigma1532]

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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°.