Simulate the number of uniformly distributed quarter circle sectors required to cover a circle; see https://math.stackexchange.com/questions/3667404.

Question3667404.java

import java.util.ArrayList;
import java.util.Iterator;
import java.util.List;
import java.util.Random;

public class Question3667404 {
    final static long ntrials = 10000000;
    final static Random random = new Random();
    
    static class Range {
        double a;
        double b;
        
        public Range(double a,double b) {
            this.a = a;
            this.b = b;
        }
        
        public boolean contains (double x) {
            return a <= x && x < b;
        }
    }
    
    public static void main(String [] args) {
        long count = 0;
        for (long trial = 0;trial < ntrials;trial++) {
            List<Range> ranges = new ArrayList<>();
            ranges.add(new Range(0.25,1));
            count++;
            while (!ranges.isEmpty()) {
                count++;
                double beg = random.nextDouble();
                double end = beg + 0.25;
                Iterator<Range> iterator = ranges.iterator();
                List<Range> newRanges = new ArrayList<>();
                while (iterator.hasNext()) {
                    Range range = iterator.next();
                    if (range.contains (beg))
                        newRanges.add(new Range(range.a,beg));
                    if (range.contains(end))
                        newRanges.add(new Range(end,range.b));
                    if (beg < range.b && end > range.a) {
                        iterator.remove();
                        ranges.addAll(newRanges);
                        break;
                    }
                }
            }
        }
        System.out.println(count / (double) ntrials);
    }
}