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 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.


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.

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. しかしながら, 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.

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.

私は月曜日に2週間ごとにここにパズルを設定します. 私はいつも素晴らしいパズルを探しています. 提案したい場合, メールを送ってください.

私はいくつかのパズルの本の著者です, つい最近 言語愛好家のパズルブック. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.