π Day

Drop random points into a square with an inscribed circle. The ratio of points inside the circle to total points, multiplied by four, converges to π.

The alternating series 1 − 1/3 + 1/5 − 1/7 + … converges to π/4. Each term nudges the sum closer, overshooting then undershooting.

Drop needles of length equal to the gap between parallel lines. The probability a needle crosses a line is 2/π, so π ≈ 2 × total / crossings.