On a circular billiard table, hit the ball so it bounces off the rail into a second ball
Two balls sit anywhere on a circular billiard table. Using only the cushion — no other balls, no pockets — where on the rail must you aim so that a ball struck from the first point bounces off the wall and lands exactly on the second? An 11th-century scholar in Cairo posed this as a problem about reflections in a curved mirror, centuries before anyone played billiards.
Reveal the answer
Ibn al-Haytham (Alhazen) solved it in his Book of Optics (~1027 CE) as a problem about light reflecting off a spherical mirror; the billiard version is mathematically identical. His construction intersects the circle with a hyperbola, reducing to a quartic equation with up to four valid bounce points depending on where the two points sit. It resisted a fully general classical compass-and-straightedge solution for centuries and is considered one of the first genuinely hard problems in the history of optics.
— Ibn al-Haytham (Alhazen), Book of Optics (Kitab al-Manazir) — c. 1027 CE, Cairo