One Shot, Twenty-One Balls: Existence and Rarity of a Total Clearance in a Single Stroke of Snooker

Abstract

Snooker folklore holds that no single stroke can pocket all twenty-one object balls. We examine the claim in an idealized but fully specified model of billiard dynamics. Within the model we exhibit an admissible configuration of the twenty-two balls and a stroke of the cue ball that pockets all twenty-one object balls, and we show that the set of such strokes has positive Lebesgue measure in the natural shot space: total clearances are not flukes of measure zero but open events. For the regulation opening configuration we conjecture the same and explain both why a simulation cannot settle the conjecture by brute force and what kind of computation could settle it in principle. Monte Carlo experiments in the same model estimate the probability P(k) that a uniformly random stroke pockets exactly k balls; the observed decay of P(k), extrapolated conditionally on the conjecture, places the probability of a total clearance from the break far beyond anything observable. The folk claim is thus right in practice and wrong in principle, and the gap between the two is exactly the distance between measure zero and unobservably small.