Mirrors and Light Puzzle
Statement of the Puzzle
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Consider an isosceles triangle ABC with apex angle A of one degree
and base BC of length 2. Let D, E bisect AB, AC, respectively, and then erase all parts of the figure except for the segments DB and
EC. Thus we have formed a truncated one-degree angle, with the distance at one opening equal to twice the distance at the other.
Imagine these two lines act as reflective mirrors, and a light ray (that lives in the plane) enters the larger opening. It will bounce
around the angle a certain number of times before exiting at the smaller opening. What is the maximum number of reflections that
are possible?
(Source: problem text from http://mathforum.org/wagon/spring02/p953.html,
illustrations and solution by Graeme McRae)
To the left is the picture that goes with the description. The ray of
light, red, enters at the lower right, and bounces back and forth several
times before exiting at the upper left. |
A different way to look at it
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Consider two concentric circles, with radius 1/sin(.5º) and 2/sin(.5º),
respectively (about 115 and 230). Now draw 360 rays from the circles' center, spaced at 1º intervals.
Finally, erase the part of each ray that is outside the larger circle or inside the smaller circle.
You're left with 360 line segments, each of length about 115. (I
just drew six here, in light blue. More would make the diagram too
"busy".) Imagine each line segment is transparent, like panes of glass glass, and a light
ray (that lives in the plane) enters from outside the outer circle. It will pass
through a certain number of panes of glass before entering the smaller circle. The light ray continues in the smaller circle, passing
"under" at least
one of the panes of glass before "petering out". What is the maximum number of panes of glass that
the ray can pass through?
The answer to this new question will be the same, because the light ray in the new question exits each transparent pane of glass at the same
angle that it would have been reflected in the original question. The advantage of the reformulated question is that simple geometry tells us
that the central angle subtended by the path of the ray of light (from where
it enters the outer circle to where it exits into the inner circle) is less
than 60º, but it could be very close to 60º, and so it can pass through as
many as 60 panes of glass, but no more.
And that's the answer to both questions.
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