Welcome to Open Problem #3! This problem is rather different than the last two, so if you haven't jumped in yet, now is a very good time to do so! You'll find artistic imagination to be just as useful as mathematics.
It's possible to design a polygonal billiard table such that there are two points that can never connect with one "shot". That is, it's possible to set up a ball in one place and a hole at another such that if the ball is hit once and reflects and unlimited number of times, it will never reach the hole.
(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.)
This has been done with a 26-sided table and a 24-sided table. However, it is unknown if 24 sides is the minimum.
Problem #1: Make a different billiard table that fulfills the above conditions, with any number of sides.
Problem #2: Either prove that 24 sides is the minimum or provide a counterexample.
Problem #1 should be considered truly open \(-\) there aren't many examples, and creativity is encouraged, even if the number of sides is high.
In addition to tackling the problems above, an excellent contribution to this project would be to make a computer program that allows playing with different configurations and seeing what happens with the reflection.
In order to make it easier to post solutions, I've rendered Problem #1 as a problem to solve in the group. Post any images you might create there.
Before you start, you should study the example below, which was the first given to show that that the no-reflections-pair setup with a polygon was possible.
It comes from a paper by George W. Tokarsky. The proof is fairly short, but requires an intermediate proof.
Lemma: Prove that, starting at the marked vertex of the billiard table shown (shaped like an isosceles triangle), it is impossible to hit a ball that reflects back to where it started (given any number of reflections).
Knowing the Lemma, this proof then follows:
If there were a pool shot from \( A_0 \) to \( A_1 ,\) the initial path must pass through the interior of one of the eight triangles surrounding \( A_0 . \) Let us call this triangle \(T.\) As in the lemma, a pool shot from \( A_0 \) to \( A_1 \) would correspond or fold up to a pool shot from \( A_0 \) to \(A_0\) of \(T,\) which is impossible.
Update: I highly recommend checking this solution from David Vreken which includes some new diagrams that work and explain a general method.
Quick update on Open Problem #2: I have rendered the text from the proof the group made into a PDF file suitable for publication, using the official American Mathematical Society format. I'm going to make a few review passes first before posting it.