A wooden floor, boards all the same width. You drop a needle exactly as long as a board is wide, and ask one question: did it land across a crack? Do it once and you learn nothing. Do it ten thousand times, divide, and a number you know very well starts to surface, from a floor, a needle, and pure chance, with not a circle in sight.
For the curious kid.
Where is the circle hiding? In the needle's spin. When it lands pointing straight along the boards it can't cross a crack at all; when it lands straight across, it crosses almost surely; in between, it depends on the tilt. Every possible tilt is a direction on a compass, and a compass is a circle. So the chance of crossing is really a question about how much of a circle's worth of directions are "crossing directions", and that answer has π in it. Drop a needle and watch the lower-left picture: each needle becomes one dot, placed by how tilted it was and how close its middle fell to a crack. The gold dots, the crossers, all sit under one smooth curve. The curve is theory; the dots are luck. The fraction of the box under the curve is 2/π, and that is where the number comes from.
It also shows something honest about luck: it is slow. The lower-right chart watches the guess for π settle as needles pile up. The band around the gold line is how wrong chance is allowed to be, and it shrinks only with the square root of the count. A hundred times more needles buys you one more decimal. Press "ten thousand" a few times and watch the guess wobble inside the band, never quite landing.
Deeper.
Georges-Louis Leclerc, Comte de Buffon, posed it in 1733 and solved it in 1777, the first problem in what we now call geometric probability. A needle of length ℓ falls on lines spaced d apart, with ℓ ≤ d. Let r be the distance from its centre to the nearest line, uniform on [0, d/2], and θ its angle to the lines, uniform on [0, π). It crosses exactly when r ≤ (ℓ/2)·sin θ. In the (θ, r) rectangle, the crossing region is the area under half a sine wave, which is ℓ, while the whole rectangle is π·d/2. So P(cross) = 2ℓ/(πd), and if you count c crossings in n drops, π ≈ 2ℓn/(cd). That rectangle is the lower-left picture, and the sine curve is drawn from the formula, never fitted to the dots.
The estimate's spread is set by the binomial: the standard error of π̂ is (2ℓ/d)/p² · √(p(1−p)/n), which for ℓ = d comes to 2.37/√n. That is the band on the right, drawn at two standard errors. It explains the famous cautionary tale: in 1901 Mario Lazzarini reported 3,408 drops with ℓ/d = 5/6 and 1,808 crossings, giving π ≈ 3.1415929, correct to six places, some hundred times better than chance would allow. His numbers produce exactly 355/113, the ancient best fraction for π, and 3,408 is a suspicious multiple of 213: with that needle, every 213 drops give the chance of hitting 355/113 dead on, and a patient man who stops the moment it happens will "measure" π to six decimals. Set the slider to 10 twelfths and you have his needle; you will not have his luck, because the exhibit does not know when to stop.
Honesty notes, museum policy. Angles and positions are uniform pseudo-random draws; a crossing is decided by the inequality above, which was checked headless against the plain geometry of the two endpoints on two hundred thousand needles with no disagreement. The sine curve and the error band come from the formulas alone. The floor's edges are harmless: the centre lands uniformly over whole board-widths, so the periodic law is exact. One more thing hides on this floor, for the patient.
Nothing here is round. The circle is in the not-knowing which way the needle would fall.