education

Did you solve it? Gods of snooker


Earlier today I set you the following puzzle:

Baize theorem

A square snooker table has three corner pockets, as below. A ball is placed at the remaining corner (bottom left). Show that there is no way you can hit the ball so that it returns to its starting position.

The arrows represent one possible shot and how it would rebound around the table.
The arrows represent one possible shot and how it would rebound around the table.

The table is a mathematical one, which means friction, damping, spin and napping do not exist. In other words, when the ball is hit, it moves in a straight line. The ball changes direction when it bounces off a cushion, with the outgoing angle equal to the incoming angle. The ball and the pockets are infinitely small (i.e. are points), and the ball does not lose momentum, so that its path can include any number of cushion bounces.

Solution

STEP 1 When a ball rebounds off a cushion, its ingoing and outgoing angles are the same. Thus if we were to consider the path of the ball continuing in the mirror image of the table, the path of the ball is a straight line, as illustrated below.

Each time the ball hits a cushion it continues into the mirror image of that table. By constructing mirror images of mirror images we get a grid of mirror images, and the straight line path of the ball continues ad infinitum (unless it falls in a pocket.)
Each time the ball hits a cushion it continues into the mirror image of that table. By constructing mirror images of mirror images we get a grid of mirror images, and the straight line path of the ball continues ad infinitum (unless it falls in a pocket.)

STEP 2 Below is a full grid of mirror images. Each square represents a table, and the red dots are the corners with no pocket. (Every other intersection has a pocket.) The only way to return the ball to its initial position would be to follow a path that is associated with a line segment that connects the bottom left corner of the grid to a red dot and does not contain any other red dots. However, as both coordinates of a red dot must be even numbers, the midpoint of such a line segment will be a grid point that corresponds to a pocket of the billiard table. Hence, it is not possible to return the ball to its initial position.

solution

Nice! I hope you enjoyed this puzzle – I’ll be back in two weeks with a new one.

Thanks to Dr Pierre Chardaire, associate professor of computing science at the University of East Anglia, who devised today’s problem.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.



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