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 6 of 7: Explicit pairs realize almost every such n

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(a,b)=(1,3k+1)\Rightarrow n=6k+1;\qquad (a,b)=(3k-1,3k+2)\Rightarrow n=12k-1

Detailed analysis

The pair $(1,3k+1)$ , $k\ge0$ , always has $\gcd=1$ and $1\equiv3k+1\pmod3$ , giving every $n=6k+1$ with $k\ge0$ . The pair $(3k-1,3k+2)$ , $k\ge1$ , has $\gcd(3k-1,3k+2)=\gcd(3k-1,3)=1$ and matching residues mod $3$ , giving every $n=12k-1\equiv11\pmod{12}$ with $k\ge1$ . For the remaining residue $n\equiv5\pmod{12}$ with $n\ge29$ , the pairs $(3k-1,3k+5)$ (for $k$ even) and $(3k-4,3k+8)$ (for $k$ odd), $k\ge2$ , are coprime with matching residues mod $3$ and give $n=12k+5$ .