MathLabs

Problem 4

Let ABCABC be an equilateral triangle. From vertex AA 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 θ\theta it leaves with directed angle 180∘−θ180^\circ-\theta. After nn bounces the ray returns to AA without ever landing on either of the other two vertices. Find all possible values of nn.
Step 3 of 7: Count crossed lines as bounces
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n=2(a+b)−3n=2(a+b)-3
Detailed analysis

The segment from A=(0,0)A=(0,0) to (a,b)∈V∩U(a,b)\in V\cap U crosses a−1a-1 lines parallel to ABAB, b−1b-1 lines parallel to ACAC, and a+b−1a+b-1 lines parallel to BCBC, each crossing corresponding to one bounce in the folded picture; the total is (a−1)+(b−1)+(a+b−1)=2(a+b)−3(a-1)+(b-1)+(a+b-1)=2(a+b)-3, so n=2(a+b)−3n=2(a+b)-3.