Impossible Bounce Problem
Consider a polygonal billiard table with two marked points, one for a "ball" and one for a "hole". Suppose the ball is "hit" once to go off in any direction and the reflects off the billiard table an unlimited number of times. Is it possible to configure this such that the ball will never reach the hole, no matter which direction the ball is aimed at?
(This assumes geometric purity: the "ball" has no mass and the "ball" and "hole" are considered single points. Also, if the ball hits a vertex point as opposed to a side then the reflections are absorbed.)
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It is impossible for a ball placed on the red dot on the right isosceles triangle billiard table pictured below to be reflected back to itself, given any number of reflections:
(See here for a proof.)
Therefore, using this right isosceles triangle as a building block, a polygonal pool table can be constructed (through a series of reflections of the building block) such that a ball can never reach the hole, as long as the hole and the starting position of the ball are on reflected red dots, and as long as all the reflected blue dots are on a corner or outside the polygonal pool table. (If a reflected blue dot is on the inside or edge of the polygonal pool table, it may be possible to make a path from the ball to the hole through that dot. See here for more details.)
Using these guidelines, there are several polygonal pool tables that can be constructed. Here is one with 24 sides (which is really just a variation of the Castro solution):
Here is one with 25 sides:
And here is one with 27 sides:
Note: While solving this I noticed that a right isosceles triangle is not the only shape that has the property that it is impossible for a ball to be placed on a certain point and reflect back on itself. A square, a rectangle, a 30-60-90 triangle, and a 60-90-120-90 kite also have this property. These are pictured below, where the red dot would be the starting point of a ball:
I am not sure if there are any other shapes with this property. Also, I was unable to make a polygonal pool table using any of these shapes as building blocks (other than the right isosceles triangle) with the given property that the ball will never reach the hole. Perhaps someone else can do this or prove that it is not possible.