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 5 of 7: n is odd and not a multiple of three

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n=2(a+b)-3\ \text{is odd},\ 3\nmid n \ \Longrightarrow\ n\equiv1\ \text{or}\ 5\pmod6

Detailed analysis

Since $n=2(a+b)-3$ is (even) minus $3$ , it is odd. An odd integer that is not a multiple of $3$ is congruent to $1$ or $5$ modulo $6$ , so every possible $n$ satisfies $n\equiv1\pmod6$ or $n\equiv5\pmod6$ .