Three-way billiards

The Workings

Today’s divertimento features the game of billiards, whose history has a vague origin. There is a widespread belief about the invention of this game, which points to a dispute between the English and the French: the former would attribute it to the British Bill Yard, and the latter to Henry Devigne, a craftsman at the court of Louis XV. These hypotheses lose weight when it is realised that the name Bill Yard refers to the garden of the Londoner William Kew (Bill’s Yard), while other sources place Henry Devigne in the time of Charles IX. What is difficult to dispute is the presence of Physics and Mathematics in billiards. For example, the article Théorie mathématique des effets du jeu de billard on the collision of the spheres and the trajectories of the balls by G. Coriolis (after whom the Coriolis effect is named), considered to be one of the first technical studies of a sport.

The Fun

Billiards is a game or sport that raises interesting questions. Today we pose the following one. We consider a billiard table, \(OABC\), where \(OC=AB=2\) and \(OA=BC=1\). We have the cue ball in O and we want to hit the red ball on three rails, i.e. the cue ball must hit the red ball after hitting three rails of the table. So, for example, we hit the cue ball with the cue stick, which must hit \(AB\), \(BC\) and \(CO\) successively and then touch the red ball.

Assuming that the balls are point balls and that the cue ball is hit without english, we ask in which region of the table the red ball can be placed so that the cue ball can reach it.

(Hint: We can set up a Cartesian system so that \(O=(0,0)\), \(A=(1,0)\), \(B=(1,2)\) and \(C=(0,2)\) and describe the region by giving its boundary.)

Solutions

We encourage the readers to try to solve the divertimento for themselves. Whether you succed or not, you can always consult the solution in this link.