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 2 of 7: Images of A and vertex-avoiding directions

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V=\{(a,b): a\equiv b\pmod3\},\qquad U=\{(a,b): \gcd(a,b)=1\}

Detailed analysis

Reflecting a vertex $(a,b)$ across a side of a unit triangle preserves $a-b\bmod3$ , and any two vertices with the same residue can be connected by such reflections, so the set $V$ of images of $A$ is exactly the lattice points with $a\equiv b\pmod3$ . A ray from $A$ toward $(a,b)$ passes through another lattice point before $(a,b)$ exactly when $\gcd(a,b)=d>1$ , since then $(a/d,b/d)$ lies on segment $A(a,b)$ ; so the ray first reaches a vertex at $(a,b)\in U$ exactly when $\gcd(a,b)=1$ . The ray returns to $A$ without hitting another original vertex first precisely for the directions toward $V\cap U$ .