You're standing in front of a square mirror that's 300 mm on a side. There's a tiled wall behind you in which the tiles are squares 150 mm on a side. When you look in the mirror, you see exactly 25 of these tiles--or rather you would, if your body didn't get in the way of some of them.

Without being told the width of the room, identify what fraction of the way from the mirror to the tiled wall that you are located.

I’m going to assume my eyeballs are a point in space. Also, I’m going to assume that my eyes are centered (left/right) on the mirror, but level with the bottom of the mirror. This is an unnecessary assumption because it doesn’t matter where my eyes are. But it makes my calculations simple.

Ok, I’m getting myself a little confused about this problem, and I’m probably making it harder that it is, but here’s how I see it. Let’s say the distance between the mirror and the tiles is D. Let’s say the distance from my eyeball to the mirror is x. So the image I see in the mirror is equivalent to if I were looking at a tiled wall through an opening (instead of the mirror) that is D+x away from me.

(Side view: tiles, then opening/mirror, then the _ is my eye)

|\
| \
|  \
|   \
|    \
|     \
|     |\
|     | \
|     |  \
|  D  | x \_

So according to my theory, there should be a straight line connecting the edge of the square of tiles to the edge of the opening (mirror) to my eye.

Well, seeing 25 tiles means I’m seeing a 5x5 square of tiles. So the total square of tiles I am seeing is 150mm*5=750mm on a side.
So the height difference between my eye and the top of the square tiles is 750mm. The horizontal distance between my eye and the top of the square tiles is D+x.
The height different between my eye and the top of the opening/mirror is 300mm. The horizontal distance between my eye and the top of the opening/mirror is x.
And the height different between the top of the square tiles and the top of the opening/mirror is (750-300)mm = 450mm. The horizontal distance between the square tiles and the top of the opening/mirror is D.

Ok, so the slope of the line connecting the edge of the square of tiles to the edge of the opening (mirror) to my eye is
M = 750/(D+x) = 300/x = 450/D.

The question is asking what fraction of the way from the mirror to the tiled wall that I am located. So it wants to know x/D. Ok, well if I use the last two representations of M I get:
300mm/x = 450mm/D
So x/D = 300/450 = 2/3.

So my distance to the mirror is two thirds the length of the room.

Posted by nikki on 2005-01-13 14:11:56