Problem 4

Let

ABC

be an equilateral triangle. From vertex

A

we draw a ray toward the interior of the triangle so that it reaches one of the sides. When the ray reaches a side it bounces off following the law of reflection, that is, if it arrives with directed angle

\theta

it leaves with directed angle

180^\circ-\theta

. After

n

bounces the ray returns to

A

without ever landing on either of the other two vertices. Find all possible values of

n

Step 1 of 7: Triangulate by unfolding reflections

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A=(0,0),\quad B=(1,0),\quad C=(0,1)

Detailed analysis

Repeatedly reflecting the equilateral triangle across the sides it bounces off tiles the plane by unit triangles. Assign each lattice vertex coordinates $(a,b)$ with $a,b\ge0$ integers, where $a$ counts unit steps in the direction of an initial side and $b$ in the direction of the other, so the starting triangle has $A=(0,0)$ , $B=(1,0)$ , $C=(0,1)$ . Every unit triangle is obtained from $ABC$ by a sequence of such reflections, and the billiard ray unfolds to the straight segment from $A$ to the image of $A$ at the point where the ray would finally arrive.