Problem 13.1.9

Solution

Let's first find a pattern in the images.

1. Let's take one mirror, and it will form $180^{\circ}$ along its horizontal, and we will get $1$ image of the object.
2. Now let's place two mirrors at an angle of $90^\circ$, and we will get $3$ images.
3. Obviously, there is a pattern, but what is it?

The pattern lies here in the angles between two mirrors.

Let's mentally draw a circle circumscribed around one mirror:

We get the following: There are $360^\circ$ in a circle altogether; a circle divided in two by a mirror having $180^\circ$ with itself, then we have the following
$n = \frac{360^\circ}{180^{\circ}} = 2$, two is the total number of objects, including the object itself in front of the mirror, so the number of images can be $N = n - 1 = 1$; also, in general, we can say the following:
$$\boxed{N = n - 1 = \frac{360^\circ}{angle \ between \ two \ mirrors} - 1 }$$
4. Let's check our formula: apply what we derived to two mirrors forming an angle of $90^\circ$

$N = n - 1 = \frac{360^\circ}{90^\circ} - 1 = 4 - 1 = 3$

Our formula works)

5. Now let's accept the fact that the prism is regular $\Rightarrow$ the angle between the mirrors is $60^\circ$, then we have:

$N = n - 1 = \frac{360^\circ}{60^\circ} - 1 = 6 - 1 = 5$

i.e. we will get 5 images of our object.

6. Conclusion:

And in general, one can look into a kaleidoscope; it consists of the prism described in the problem statement, and the only differences may be in the angles.