Nonplussed! by Julian Havil Page A

must conclude that he was a patient man.
    Of course, it is that penultimate entry which stands out, initially because of the curious 3408 repetitions (of a needle with l = 2.5 cm tossed across parallel lines with d = 3 cm). What also stands out is that 3.141 592 9 is the seven-decimal-place approximation to the second-best-known rational approximation of π of(known in the fifth century to Tsu Chung-chih).
    Perhaps it was deception. If we follow Badger’s and O’Beirne’s reverse engineering, since 2 l /( πd ) ≈ R / C it is the case that2 Cl / dR ≈ π and if we use our rational estimate for π we have that

    Table 7.3. Empirical compared with theoretical data.

    A reasonable choice is 2 l = 5, which makesand since d > l a reasonable choice for that is d = 3 and this makes C / R = 213 k /113 k . Provided that C and R are chosen to make their ratio, the result will be achieved: with k = 16 we have Lazzarini’s figures.
    Or it may have been luck. With 2 l / πd ≈ R / C andwe have that 5/3 π ≈ R / C and so π ≈ 5 C /3 R . O’Beirne points out that one trial earlier than the given final repetition of R = 3408 would have R = 3407 and C = 1807 or C = 1808, which would make the estimate π ≈ 3.142 … and π ≈ 3.140 …, respectively, each out in the third decimal place. In turn, Badger points out that had there been C = 1807 or C = 1809 crossings in 3408 repetitions, the estimates would be π ≈ 3.143 … and π ≈ 3.139 …, respectively; the experiment does seem to have stopped on something of a cusp of luck.
    Now consider the rest of the data. Again, with, the probability of a crossing is 2 l / πd = 5/3 π and so, on average, the expected number of crossings is (5/3 π )× R and if we extend table 7.2 to include these values we arrive at table 7.3 .

    Figure 7.10. Convexity and nonconvexity
    It all looks too accurate and some simple statistical tests quantify that suspicion; the chances of this happening are less than 3 × 10 −5 . And this is our, but by no means Badger’s, final word on the topic.
    A Generalization and a Final Surprise
    Buffon mentioned throwing a square object: in fact, we can formulate a surprising result for any convex, polygonal lamina. First, a polygonal lamina is convex if it contains all line segments connecting all pairs of points on it. For example, figure 7.10 shows a convex and a nonconvex pentagon.
    Note that an immediate consequence of convexity is that any straight line will intersect precisely two sides of the lamina or none at all.
    Now suppose that we throw the lamina onto a set of parallel, ruled lines a constant distance d apart. Suppose also that the lamina is made up of n sides of length l i for i = 1, 2,…, n , where each side is less than d . Since the order in which we count the sides is irrelevant, the intersection of a ruled line with the lamina must occur with the line pair ( l i l j ) of the lamina for some pair i and j , where we may assume i < j , and suppose that this occurs with probability P ( l i l j ). This means that the probability of an intersection of the lamina with a line is
    If the side l i is intersected with probability P ( l i ), since the lamina is convex, so must exactly one of the remaining sides, and soThis means that

    Now we use the previous result for Buffon’s Needle to write P ( l i ) = 2 l i / πd and sowhich makes

    the probability that the lamina crosses a line is completely independent of its shape, depending only on its perimeter.
    There are many more variants and generalizations of the original, novel idea of the eighteenth-century polymath Georges Louis Leclerc, Comte de Buffon: instead of parallel lines a rectangular grid, or perhaps radial lines or unequally spaced lines with a needle with a ‘preferred’ orientation (which is apparently useful in determining the spacing of flight lines for locating anomalies in airborne geophysical surveys). The Monte Carlo technique, of which this is the original example, is commonly