The model

Published (Buffon 1777, Laplace 1812): a needle of length l dropped on lines d apart, with l ≤ d, crosses a line with probability 2l/(πd). With Lazzarini's l = 2.5 cm and d = 3 cm that is 5/(3π) = 0.5305, so π ≈ (5/3) × throws / crossings. Published (Lazzarini 1901): 1,808 crossings in 3,408 throws, which gives 355/113 exactly.

Ours: each throw on this page drops the centre of the stick uniformly on the floor and points it in a direction taken from a random point in a circle, so no π goes in. The honest rule decides on 3,408 throws before the first. Lazzarini's rule, which is our reading of how 355/113 could be reached honestly, looks at the count after every 213 throws and stops the first time the crossings are exactly 113 for each 213. The exact column is computed in Python, by the binomial distribution and by dynamic programming over the 16 looks.