# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1039: Mirror, mirror

From New Scientist #2195, 17th July 1999 [link]

George has fitted two mirrors inside a rectangular box, the first on one of the long sides, and the other across a diagonal. He shines a narrow beam of light through a small hole which he has drilled through the side of the box and the mirror on that side. It enters perpendicular to the side of the box, is reflected five times by the mirrors and emerges perpendicular to the end of the box through another hole, as shown.

The box is exactly one metre long. You can calculate the width if you wish, but that is not George’s problem. He wants to know how far the beam travels while it is inside the box.

Can you help him?

[enigma1039]

### One response to “Enigma 1039: Mirror, mirror”

1. Jim Randell 26 October 2018 at 7:24 am

As in Enigma 1532 (and, more recently, Teaser 2503) it is easier to think about the light beam travelling in a straight line and reflect everything else at the mirrors.

There are 6 segments of the light beam in the box, and the final segment has turned through 90° compared to the first segment. This means that angle between the mirrors is 90° / 6 = 15° and the reflected box associated with the final segment is at 90° to the original box.

Here is a diagram with the light beam drawn as a straight line, the positions of the reflected mirrors and the final reflection of the box shown:

From which we see that the distance the light beam travels inside the box is the same as the length of the box.

Solution: The beam of light travels 1m inside the box.

The width of the box is 1m × tan(15°), approximately 268mm.

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