The Pool Table Problem
On a rectangular table with whole-number sides, which corner does a bounced ball reach, and after how many bounces? Use the reflection trick to turn the path into a straight line and prove the pattern.
Where this idea comes from
Start here — these are the sources that inspired this exploration.
Introduction
A ball is struck across a rectangular billiard table with no pockets except the four corners, and bounces off the cushions at equal angles until it reaches a corner. On a table with whole-number side lengths, the ball always eventually reaches a corner, and the number of bounces and the corner it reaches both follow a pattern based on the table's dimensions. This exploration uses the 'unfolding' trick - reflecting the table repeatedly instead of the ball - to turn the bouncing path into a single straight line, and asks you to find and prove the pattern.
Guiding Questions
- A ball is hit around a billiard table and you want it to return to its start. What paths make this happen?
- Use the unfolding (reflection) trick to turn a bouncing path into a straight line. What does it tell you?
- On a table with whole-number side lengths, when does the ball end in a corner, and after how many bounces?
- What patterns appear for different table proportions, and can you prove one of them?