Buffon's Needle: π from the Floorboards
About this app
Needles fall onto floorboards and reveal π. The proof without integrals goes via pieces and a noodle to a circle that always crosses the joints exactly twice.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What is Buffon’s needle problem?
In 1777 Georges-Louis Leclerc de Buffon asked: a needle of length ℓ falls at random onto a floor of boards whose joints are d apart. How likely does it lie across a joint? For ℓ ≤ d the answer is 2ℓ/(πd). So the circle constant π appears in a random experiment with straight needles. Throw N needles, count T hits, and 2ℓN/(dT) estimates π. It approaches π only slowly, though: the error shrinks roughly like 1/√N.
The proof without integrals
In 1860 Joseph-Émile Barbier counted crossings instead of hits. The mean number of crossings is additive, even when two pieces are rigidly joined, because expectation is linear. So it is proportional to the length, for every shape, including a bent needle. A circle of diameter d always crosses the joints exactly twice and has circumference πd. From c·πd = 2 we get c = 2/(πd). A short straight needle crosses at most once, so its mean is the hit probability 2ℓ/(πd).
What you try in the app
The app walks through the proof in seven steps: one needle falls, then a needle rain with glowing hits and a running π estimate, a convergence curve with a 1/√N funnel, a bent needle, a noodle and a circle you can drag across the floor. Histograms show that a straight needle, a noodle and a circle of the same length cross equally often on average. In free mode you choose shape, length, joint spacing and number of throws; fixed, numbered randomness makes every result repeatable.
Frequently asked questions
- How do you calculate π with Buffon’s needle?
- Throw N needles of length ℓ onto boards with joint spacing d (ℓ ≤ d) and count the T needles that hit a joint. Then π ≈ 2ℓN/(dT). For ℓ = d this simplifies to π ≈ 2N/T.
- Why does π come out of straight needles?
- The mean number of crossings depends only on the length, not on the shape. A circle of diameter d has circumference πd and always crosses exactly twice. Hence any length ℓ gives 2ℓ/(πd) crossings on average.
- How accurate is the estimate after 1000 throws?
- Usually only one or two decimals are right. The typical error shrinks like 1/√N: one more digit needs about a hundred times as many throws. The experiment is a beautiful proof but a slow way to compute.
- Can I throw long or bent needles in the app?
- Yes. In the Explore panel you choose straight, zigzag, arc or circle and lengths up to 30 cm. For long or bent needles the app shows the mean number of crossings, which is still 2ℓ/(πd).
Subject: From Erdős's BOOK
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