In 1777 the Comte de Buffon asked a question that sounds like idle amusement: throw a needle onto a floor of evenly spaced boards, and how often will it come to rest across a seam? The answer is not idle at all. When the needle is exactly as long as a board is wide, it lands across a seam with probability 2 ⁄ π. So count the crossings, divide, and the circle-constant falls out of a heap of dropped iron.
Nothing on this floor is drawn from π. Each needle arrives at a random place and a random tilt; the estimate is only ever two whole numbers — needles thrown, and seams crossed. Early on it lurches wildly. Then the Law of Large Numbers tightens its grip and the error closes like 1 ⁄ √N: to buy one more honest decimal you must throw a hundred times as many needles. The last digits are ruinously expensive.
Watch the ember needles — those are the crossings, the ones that pay. The muted gold ones fall between the seams and pay nothing, yet they matter just as much: π lives in the ratio, not in either count alone. Scatter a handful of your own and the estimate flinches, then settles. Chance, piled deep enough, stops being chaos and becomes a constant.