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More bouncing (Posted on 2016-03-09) Difficulty: 3 of 5
Place an isosceles right triangle with its right angle at the origin and legs along the positive x- and y-axes.

Imagine this triangle is three mirrors and a laser is fired from the origin with some positive slope.

The laser will bounce around and one of 3 possibilities will occur:
1. The laser will eventually come back to the origin.
2. The laser will eventually hit one of the other corners.
3. The laser will never hit any of the corners.

For a given slope, how can you tell which will happen?

No Solution Yet Submitted by Jer    
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Solution proposed solution | Comment 1 of 3
As in Light Beam Reflection, a tesselation of the plane will allow treating the light beam as one straight ray.

The triangle is reflected about each side to form extra triangles, and this process continues with each successive triangle so created.

Consider the original triangle to have legs of unit length, with a hypotenuse of sqrt(2).  Each 2x2 block of the tesselation will contain 8 triangles; or you can consider it divided into 4 squares, overlaid with a diagonally oriented square with sides of length sqrt(2).

If the slope is irrational, the beam will never hit a corner. If the slope is rational, reduce it to the lowest possible terms, such as 1/2, 1/3, 7/5, 8/7, etc.

It looks, from a drawing on graph paper, that, if the numerator or denominator is a multiple of 2, then the beam eventually hits a different corner from the origin. If neither the numerator nor denominator is a multiple of 2, then it returns to the origin.

Edited on March 10, 2016, 7:34 am
  Posted by Charlie on 2016-03-09 14:28:48

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