One Shot, Twenty-One Balls: Existence and Rarity of a Total Clearance in a Single Stroke of Snooker
2026-07-14 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors investigate a popular snooker belief that no single shot can pocket all twenty-one balls. Using a detailed billiard model, they show it's actually possible to make such a shot and that these shots are not just rare flukes but occur within a measurable set of possibilities. However, for the standard starting layout of the balls, they think it’s still nearly impossible to do in practice, which explains why simulations can’t easily prove or disprove this. Their experiments suggest the chance of clearing all balls in one shot during a normal break is incredibly small—effectively unobservable. So, the traditional belief is mostly true when playing but not strictly true in theory.
snookerbilliard dynamicscue ball stroketotal clearanceLebesgue measureMonte Carlo simulationprobabilityopening configurationmeasure zero
Authors
Avner Kantor
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.