Earlier today I set you a puzzle about a mirror:

*A man is facing a mirror hanging on a wall 1m in front of him. *

*In order for him to see himself entirely in the mirror, what is the minimum possible height of the mirror and where should it be mounted on the wall?**He takes a pace backwards so he is 2m from the wall. What is the minimum possible height that the mirror now needs to be so he can see himself completely? Is the mirror larger or smaller than the**one before?*

(The puzzle requires some basic physics, which I will spell out here. First, light travels in a straight line. Secondly, when light reflects in a mirror the angle of incidence is equal to the angle of reflection. In other words, light hitting a mirror behaves just like a snooker ball bouncing against the side cushion: the angle it hits the cushion is the same as the angle of the rebound.)

**Solution**

Let’s sketch out the solution. Here’s a person standing in front of a wall. Lets begin by saying that the mirror covers all of the wall. In order for the person to see their feet in the mirror, the light from their feet must hit the mirror and rebound into the eye. Let’s say this light hits the mirror at P.

The person is standing straight, so we can assume that their feet and their eyes are the same distance from the wall. Because the angle of incidence and reflection are equal, the distance (L) from P to the eye-line must be the same as the distance from P to the floor.

We can use the same logic for what is happening above the eye-line. For the person to see the top of their head, the light must reflect at point Q, equidistant between the height of the top of their head, and their eye-line.

The mirror must contain both points P and Q and everything in between, which means the minimum height of the mirror is the distance between P and Q. And which means it has the length L + *l*. Since the person’s height is 2L + 2*l*, the mirror is exactly half the person’s size. And it needs to be mounted between points P and Q on the wall.

Nowhere in this calculation did I mention the distance from the wall. That’s because it is irrelevant to the calculation. In other words, it does not matter how far back you are when looking at the mirror. Whether you are 1m, 2m or even 1km back, the mirror only needs to be half your height for you to see yourself in it completely. Looking good!

I hope you enjoyed the puzzle. I’ll be back in two weeks.

*I set a puzzle here every two weeks on a Monday.** **I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja. I also co-write the children’s book series Football School.*