Estimating π by dropping needles on the floor

Drop a needle onto a lined floor 3,408 times, count how often it crosses a line, and you can estimate π correct to six decimal places. That, at least, is what the Italian mathematician Mario Lazzarini reported in 1901. The claim rests on a genuine and beautiful result from 1777, when Georges-Louis Leclerc, Comte de Buffon, posed what is now regarded as the first problem in geometric probability [2]. In this paper we state Buffon's result, turn it into a machine for estimating π, and then do what this series does best: check the famous numbers for ourselves. Lazzarini's experiment, we will see, is a little too perfect.

From floorboards to a π-machine

Rule a floor with parallel lines a distance $t$ apart — think of the gaps between floorboards — and drop a needle of length $l$ onto it completely at random, with $l \le t$ so the needle can cross at most one line. Buffon showed that the probability of the needle crossing a line is

$$P(\text{cross}) = \frac{2l}{\pi t}$$

Deriving this properly needs a little calculus (the needle lands at a random angle, and averaging over all angles is what drags the circle — and hence π — into the answer), so we will take the result as given and sanity-check it instead. It says a longer needle crosses more often, and more widely spaced lines are crossed less often — both obviously right. And for a needle half as long as the gap, $l = \frac{t}{2}$, it predicts

$$P(\text{cross}) = \frac{2 \times \frac{t}{2}}{\pi t} = \frac{1}{\pi} \approx 0.318$$

about a 32% chance — a claim you can genuinely test tonight with a box of matches and some ruled paper.

Rearranging the formula gives $\pi = \frac{2l}{tP}$. We can never know the probability $P$ exactly, but we can estimate it: throw the needle $N$ times, count $C$ crossings, and use $P \approx \frac{C}{N}$. Substituting:

$$\pi \approx \frac{2lN}{tC}$$

With $l = \frac{t}{2}$ this simplifies to the delightful recipe $\pi \approx \frac{N}{C}$: throw, count, divide. This is a Monte Carlo method — using randomness to estimate a quantity — the same idea used today to price financial options and simulate nuclear reactors [3].

The catch is speed. Random noise in $C$ only shrinks like $\frac{1}{\sqrt{N}}$, so each extra decimal place of accuracy costs roughly $100$ times more throws. Keep that in mind for what comes next.

Lazzarini's suspiciously perfect experiment

In 1901 Lazzarini reported using a needle of $2.5$ cm on lines $3$ cm apart (so $\frac{l}{t} = \frac{5}{6}$), throwing it $3,408$ times, and observing $1,808$ crossings. Plugging in:

$$\pi \approx \frac{2 \times \frac{5}{6} \times 3408}{1808} = \frac{355}{113} \approx 3.1415929$$

The true value is $3.14159265\ldots$, so his error is under $0.0000003$. Let's apply some honest scepticism:

1. $\frac{355}{113}$ is not a random-looking fraction — it is the most famous rational approximation to π, known to the Chinese mathematician Zu Chongzhi in the 5th century.
2. With $\frac{l}{t} = \frac{5}{6}$, the estimate is $\frac{5}{3} \times \frac{N}{C}$. To land on $\frac{355}{113}$ exactly, you need $\frac{N}{C} = \frac{213}{113}$. And behold: $3408 = 16 \times 213$ and $1808 = 16 \times 113$.
3. The result is absurdly fragile. One extra crossing ($1,809$) gives $3.1399$; one fewer gives $3.1433$. A single needle changes the error by a factor of about $6{,}500$.
4. The expected number of crossings in $3,408$ throws is almost exactly $1,808$, but with a typical random wobble of about $\pm 29$. The chance of hitting $1,808$ on the nose is roughly $1.4\%$ — a 1-in-70 shot.

The statistician Lee Badger analysed the experiment in 1994 and concluded that Lazzarini almost certainly rigged it [1] — most likely by choosing throw counts in multiples of $213$ and stopping at a convenient moment, if the throws happened at all.

Four million years of needle-dropping

Buffon's needle is entirely real: the formula is correct, the method genuinely works, and running it is a lovely afternoon experiment. But it is honest about its slowness — to earn six decimal places you would need roughly $10^{14}$ throws, which at one per second is about four million years of needle-dropping. Probability has a habit of humbling our intuition (see the birthday paradox), but it also polices fraud: when a result looks too good to be true, the same formulas that make it possible tell you it probably isn't. Lazzarini's needle didn't find π — but checking his claim is a better lesson than his experiment ever was.

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References:

[1] Lee Badger, "Lazzarini's Lucky Approximation of π", Mathematics Magazine, 67(2) (1994), 83–91.

[2] Georges-Louis Leclerc, Comte de Buffon, "Essai d'arithmétique morale", Supplément à l'Histoire Naturelle, vol. 4 (1777).

[3] Charles M. Grinstead and J. Laurie Snell, Introduction to Probability (American Mathematical Society, 1997).

Note: Our analysis assumes the needle is no longer than the line spacing and that throws are genuinely random — both harder to achieve with real needles than the mathematics suggests.