Lazzarini's Lucky Approximation of π

In 1812 Laplace [14] remarked that one could approximate Ir by performing a Buffon needle experiment. Since then several needle casters claim to have done just that. Lazzarini's 1901 Buffon approximation of rn [15] was accurate to six decimal places. This work was commended in several publications for illustrating the connectedness of mathematics [13] and validating the laws of probability [3], [7]. However, the 1960 study of Gridgeman [9] suggested that Lazzarini's experiment was not carried out in an entirely legitimate fashion and perhaps didn't wairant the praise it later received. But Gridgeman stopped short of establishing that the experiment was contrived to achieve the desired numerical result. I will begin by reviewing the history of Lazzarini's experiment and the work of Gridgeman that debunked it. I will then extend Gridgeman's work to virtually rule out any possibility that Lazzarini performed a valid experiment. Some of this work was anticipated by that of O'Beime [16], however, it also goes beyond that of O'Beime. In this study elementary applications of probability, recurrence relations, and various numerical techniques are used to look deeper into a small piece of the history of mathematics. For brief expositions of the work of Gridgeman and O'Beime see also Pilton [17] and Zaydel [18].