Nonplussed! by Julian Havil

25(3):183–95) by N. T. Gridgeman. This is reproduced in table 7.1 , with the relative length of the needle and gap equal to l / d .
    Table 7.1. The number of repetitions is R , the number of crossings is C , and the estimated value of π .

    Figure 7.5. The needle crossing a line
    With all of this experimental data, it is time to look into the mathematics of all of this.
    Figure 7.5 shows the needle crossing one of the horizontal lines at an angle θ to the positive x -direction. If we define y to be the distance of the lower end of the needle from the line which has been crossed, it must be thatand alsoThe vertical distance of the lower end to the upper end of the needle is l sin θ and for the needle to cross the line it must be that l sin θ > y . figure 7.6 shows a plot of the rectangular ‘phase space’ for the experiment, together with the curve y = l sin θ : crossings are achieved at all points underneath and on the curve.
    To calculate the probability of a crossing we need to calculate the fraction

    Figure 7.6. The experiment’s phase space
    and so arrive at the remarkable fact that

    and hence at an experimental method of approximating π .
    If we revert to the empirical, for a given length of needle and distance between parallel lines, we can perform the experiment repeatedly in the manner of our Victorian forbears (or get a computer random number generator to do the work for us) to compute the value C / R .
    In fact, ‘throwing the needle’ 10 000 times with l = 1 and d = 2 led to the result

    which, of course, means that π ≈ 3.143 17 ….
    Buffon’s (Long) Needle
    The condition thatensures that l sinand therefore that the curve lies within the rectangle in figure 7.6 . If we wish to conduct the experiment with l > d , l sin θ may well be greater than d and we will need to take into account the overlap of the curve and the rectangle, as shown in figure 7.7 , and compute the area under the truncated curve.

    Figure 7.7. The modified phase space
    The intersections are where l sin θ = d , or = sin −1 ( d / l ), and π − sin −1 ( d / l ). The area we want is then

    where the cos(sin −1 ( d / l )) is transformed to the more convenientby use of the standard mechanism that if θ = sin −1 ( d / l ), sin θ = d / l and so the triangle shown in figure 7.8 exists and the third side is found by using Pythagoras’s Theorem, which makes cos
    And all of this makes the probability of a crossing at least one line the rather more impressive expression

    Figure 7.8.

    Figure 7.9. The full story
    To summarize, the probability, P N , of the needle crossing at least one line is given by

    Notice that, not unreasonably, the two formulae agree at l = d . figure 7.9 is a plot of this combined probability function against l / d .
    The Lazzarini Entry
    The fifth entry of table 7.1 stands out. The final column would have us believe that π has been estimated to an accuracy of sixdecimal places by the method, far in excess of the accuracy of the other entries: was it luck or deception?
    Table 7.2. Lazzarini’s data.

    In 1901 the Italian mathematician Mario Lazzarini published the result under the rather wordy title, ‘Un applicazione del calcolo della probabilità alla ricerca sperimentale di un valor approssimato di π ’, in the journal Periodico di Matematica 4:140–43; four pages of fame which has led to many more pages of suspicion and of outright rebuttal. Gridgeman’s article provided compelling reasons to doubt, the excellent 1965 book Puzzles and Paradoxes by Tim O’Beirne built on that doubt and Lee Badger’s analysis in ‘Lazzarini’s lucky approximation of π ’ ( Mathematics Magazine 67(2), April 1994) pretty much signed the intellectual death warrant.
    We will not attempt to discuss the matter at any length here, but a few details from these studies are hard to ignore. In fact, Lazzarini reported the data as part of a table of results of a number of such experiments, shown as table 7.2 . We