Two plane mirrors form an angle $0^{\circ}<\alpha<180^{\circ}$ with each other. A light ray can be reflected multiple times from these mirrors. Given $\alpha$, what is the maximum number of reflections?
We think of mirrors as (infinite) half-planes with a common boundary line - edge; the light is reflected back into the interior of the domain of the angle $\alpha$ of the sheet. We do not consider physical aspects (energy loss, etc.).

The path - trajectory - of the light ray $f_{0}$ before the first reflection is perpendicular to the edge $e$ of the mirror pair. (See the note below.) In this conception, the phenomenon takes place in the plane $S$ which passes through $f_{0}$ and is perpendicular to $e$, and simplifies to reflections on the angle-bisecting half-lines $E T$ and $E U$. We denote by $V_{1}, V_{2}, \ldots$ the successive reflection points of the light ray (alternating between $E T$ and $E U$), and by $m$ the maximum number of reflections for a given $\alpha$.

![](https://cdn.mathpix.com/cropped/2024_05_02_efbe15a17405e8f8ac5cg-1.jpg?height=992&width=1352&top_left_y=455&top_left_x=381)

We call - deviating from the physical convention - the angle of incidence of $f_{0}$ the angle $T V_{1} F_{0}=\varphi_{1}$ (where $F_{0}$ is a point on $f_{0}$ before $V_{1}$). For this, $0 < \varphi_{1} < \alpha$. The light ray $f_{1}$ is reflected at $V_{1}$, and the angle of reflection is also $\varphi_{1}$. The next reflection point $V_{2}$ is on $E U$, and the angle of incidence is $\alpha - \varphi_{1}$. The light ray $f_{2}$ is reflected at $V_{2}$, and the angle of reflection is also $\alpha - \varphi_{1}$. The next reflection point $V_{3}$ is on $E T$, and the angle of incidence is $2\alpha - \varphi_{1}$. This process continues, and the $k$-th reflection point $V_{k}$ is on $E T$ if $k$ is odd and on $E U$ if $k$ is even. The angle of incidence at $V_{k}$ is $(k-1)\alpha - \varphi_{1}$, and the angle of reflection is also $(k-1)\alpha - \varphi_{1}$.

The maximum number of reflections $m$ is the largest integer such that $(m-1)\alpha - \varphi_{1} < \alpha$, but $m\alpha - \varphi_{1} \geq 180^{\circ}$. This is because for any $\varphi_{1}$, $m\alpha \geq 180^{\circ}$, which implies $\frac{180^{\circ}}{\alpha} \leq m$. In words: if $\frac{180^{\circ}}{\alpha}$ is an integer, then this is the maximum number of reflections; otherwise, it is the larger of the two integers surrounding $\frac{180^{\circ}}{\alpha}$.

Knowing the properties of the floor function $[\cdot]$, our result can be written in formula form as:

$$
m=-\left[-\frac{180^{\circ}}{\alpha}\right]
$$

(This brings us closer to the conventional interpretation of the steps at the end of our graph, especially in the case of the "bubbles" that appear in unusual situations.)

Notes. 1. With the introduced simplification, we essentially examined the (perpendicular) projection of every possible light ray path onto the plane perpendicular to the edge $e$. This was sufficient to answer the question because a spatial figure exists (is representable) if and only if its projection exists.

A concern may arise: in our projection on $S$, the angles used do not appear in their true size (they appear larger) if $f_{0}$ is not perpendicular to $e$; are our considerations then valid? Yes, they are valid. For example, the angles of $\varphi_{1}$ are always equal, and according to the part of the law not yet cited, $f_{1}$ "remains" in the plane determined by the perpendicular to the first mirror plane at $V_{1}$ and $f_{0}$. Therefore, our projection shows the projections of the two equal angles as equal.

In any case, the angle $\varphi_{1}$ played only a supporting role in our solution; it was useful because the question also involves the angle $\alpha$. It is easier to draw the path of the light ray by starting from the mirror image $F_{0}^{*}$ of $F_{0}$ on the first mirror (or $E T$) through $V_{1}$, then taking $V_{1}$ as the mirror image $V_{1}^{*}$ on the second mirror ($E U$), and directing it towards $V_{2}$, and so on. (The images $V_{1}^{*}, V_{3}^{*}, \ldots$ and $V_{2}^{*}, V_{4}^{*}, \ldots$ line up on the mirror image of $E T$ on $E U$, and the mirror image of $E U$ on $E T$, respectively.)

2. Essentially, we assumed the light ray came from infinity, but this does not change the sought number. It is possible to achieve $\varphi_{1} > \alpha$ with a light source between the two mirrors and an appropriate distance, but this already results in fewer reflections.