一杆清台：斯诺克单杆全清的存在性与稀有性
One Shot, Twenty-One Balls: Existence and Rarity of a Total Clearance in a Single Stroke of Snooker
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中文总结 AI 辅助
研究斯诺克单杆全清问题，在理想化台球动力学模型中展示全清击球方式及正测度，对标准开球配置作相同猜想，解释模拟无法解决猜想原因，通过蒙特卡罗实验估计进球概率，表明民间说法实践正确理论错误。
中文摘要 AI 辅助
斯诺克界流传着无法一杆打进全部21个目标球的说法。我们在理想化但完全指定的台球动力学模型中研究此说法。在模型中，我们展示了22个球的一种可允许配置以及能打进所有21个目标球的主球击球方式，且表明此类击球方式在自然击球空间中具有正勒贝格测度，即全清并非零测度的侥幸事件而是开放事件。对于标准开球配置，我们作出相同猜想，并解释了为何模拟无法通过暴力方法解决该猜想以及原则上何种计算可解决它。蒙特卡罗实验估计了均匀随机击球恰好打进k个球的概率P(k)；根据猜想推断，观察到的P(k)衰减表明开球全清的概率远超出可观测范围。因此，民间说法在实践中正确但在理论上错误，两者差距正是零测度与不可观测的小概率之间的距离。
英文摘要
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.
