# Sunday Times Teaser 2503 – No Title

### by Danny Roth

#### Published September 12 2010 (link)

George has placed two vertical mirrors touching each other, with an angle between them. He has also placed a small cube between the mirrors and counted how many images there are of it in the mirrors. (For example, if the mirrors had 90 degrees between them, there would be three images.) He wrote down two whole numbers — the angle between the mirrors, in degrees, and the number of images of the cube. When Martha saw the two numbers, she commented that their product, appropriately, was a perfect cube.

What was the angle between the mirrors?

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The details of my Python solution for this teaser are presented here.   On this page I am only giving details of possible solutions with some discussion of them.

In fact this teaser is not well specified and this makes it impossible to say that there is a correct solution.  In considering increasing cubes it is straightforward to eliminate those below the first possible solution $$6^3$$, giving 2 images with a mirror angle of $$128^{\circ}$$ as shown in the diagram above. This is a good candidate since the two images appear for any observer position between the mirrors.

The next possible answer is for the cube $$7^3$$ given by 7 images and a mirror angle of $$49^{\circ}$$. This is the ‘official’ answer but if we set this up with the object and the observer on the centre-line as shown above, we find that there are only 6 images!

But this does not mean that the official answer is wrong. If we change the sight-line as shown above, we then see seven images (in the right hand diagram, the distance from the object to the observer has been reduced to show all the ray paths between the two).

So the ‘official’ solution works but only when the observer and/or the object are not on the centre-line!

So what is going on here?   It turns out that a new image is added when a ray between the object and the observer passes very close to the point where the mirrors meet (lets call this the vertex).  If this potential ray hits one of the mirrors (the one to the right in our case) the angles are such that the ray would bounce between the two mirrors while moving ever closer to the vertex and disappear never to return!    If, however this potential ray hits the other mirror (the left one for us) there is now a path as shown below on which the ray can change direction and escape ‘capture’ by the mirrors and go on to create a new image.

Of course in a practical setup the point where the mirrors meet would not be perfect so we would not see this new image until the angle was such that the reflections involved were further away from the vertex.  And if the mirrors were perfect, once a ray gets within microns of the vertex we would have to give up our view of light rays as streams of photons and consider them as waves that set up a static wave field at the vertex. But I stopped at this point!

Jim Randell has also covered this teaser here where he shows the results of practical experiments with real mirrors set up to show various solutions.