To solve this problem, we need to analyze the trajectory of light rays inside a sphere and determine the number of reflections it takes for the light ray to return to its origin.

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Step 1: Understanding the geometry and reflections

• The sphere's surface behaves like a perfect reflective mirror.
• If the laser is directed at an angle $\alpha$ with the tangent to the inner surface of the sphere, the angle remains constant upon each reflection because of the law of reflection.
• The path of the laser beam forms a regular polygon inscribed in the sphere, where each side represents one reflection.

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Step 2: Key Observations

1. The total angle subtended by the laser ray at the center of the sphere after one complete round trip must equal $360^\circ$ for the beam to return to its origin.
2. Each reflection causes the laser beam to subtend an angle of $2\alpha$ at the sphere's center.

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Step 3: Finding the number of reflections ($N$)

For the laser to return to its starting point: $N \cdot 2\alpha = 360^\circ$ $N = \frac{360^\circ}{2\alpha}$

• For $\alpha = 60^\circ$: $N = \frac{360^\circ}{2 \cdot 60^\circ} = 3$ This implies the laser completes the round trip in 2 reflections (since the third point is the starting point).

• For $\alpha = 20^\circ$: $N = \frac{360^\circ}{2 \cdot 20^\circ} = 9$ This means the laser completes the round trip in 8 reflections (since the ninth point is the starting point).

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Step 4: Conclusion

For $\alpha = 20^\circ$, the minimum number of reflections the light takes to return to Emon is: $\boxed{8}$

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Would you like a detailed diagram or further clarifications? Let me know!

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Related Questions

1. What is the trajectory of the beam for other values of $\alpha$?
2. How does the law of reflection govern the beam's path inside the sphere?
3. Can the angle $\alpha$ be such that the light never returns to its origin?
4. What happens if $\alpha$ is irrational (e.g., $25.7^\circ$)?
5. How does the size of the sphere affect the reflections?

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Tip:

Visualizing this problem with diagrams and inscribed polygons helps to better understand the relationship between the angle $\alpha$ and the number of reflections.