Indeed, the result of the experiment is strongly dependent on arithmetic properties of the parameters - the ratio $\rho$ of needle's lenght to the distance between parallel lines, and of course, the number $N$ of needless. There is a funny story about that. In 1901 the Italian mathematician Mario Lazzarini claimed to have obtained an experimantal value of $\pi$ with 7 correct digits, using 3408 needles, and a ratio $\rho=5/6$. That was quite embarrassing, as it was quite clear he had cheated -after and before Lozzerini's experiment, everybody else, even with larger numbers of needles, never got a better result than 3.13 or 3.15., and today Lazzerini's experiment it is sometimes reported as a case of false. The point is a little more subtle: with the parameters he had taken, he was quite likely to obtain as approximantion the ratio 355/113 (the lucky number being 1808 intersections). My personal guess is that he somehow meant to mock the other scientists, physicists or naturalists, who had ignored the arithmetic axpect of the matter. One can easily do even more striking experiments, of course: with a convenient choice of the ratio, and just 2 or 3 needles, you have a good chance to obtain an exact value of $\pi$ from Buffon's experiment...

Indeed, the result of the experiment is strongly dependent on arithmetic properties of the parameters - the ratio $\rho$ of needle's lenght to the distance between parallel lines, and of course, the number $N$ of needles. There is a funny story about that. In 1901 the Italian mathematician Mario Lazzarini claimed to have obtained an experimental value of $\pi$ with 7 correct digits, using 3408 needles, and a ratio $\rho=5/6$ (a harmless choice, apparently). That was quite embarrassing, as it was quite clear he had cheated -after and before Lozzerini's experiment, everybody else, even with larger numbers of needles, never got a better result than 3.13 or 3.15., and today Lazzerini's experiment it is sometimes reported as a case of false. The point is a little more subtle: with the parameters he had taken, he was quite likely to obtain as approximantion the ratio 355/113 (the lucky number being 1808 intersections). My personal guess is that he somehow meant to mock the other scientists, physicists or naturalists, who performed Buffon's experiment ignoring the arithmetic axpect of the matter. One can easily do even more striking experiments, of course: with a convenient (irrational) choice of the ratio, and just 2 or 3 needles, you have a good chance to obtain an exact value of $\pi$ from Buffon's experiment...