[metaphyze]

Brute force gives me about 2287/2288. Trying each number 1,000,000 times (distribute X balls in 365 bins, count number times all bins are filled) so it seems reliable, but maybe I have bug somewhere:

import java.util.Random;

public class Birthday {

private static Random rand = new Random(System.currentTimeMillis());

public static void main(String... args) {

		final int n = 365;
		boolean[] bins = new boolean[n];

		final float maxNumberOfAttempts = 1000000;
		for (int numberOfBalls = 2285; numberOfBalls <= 2400; ++numberOfBalls) {

			int attempts = 0;
			int numberOfTimesAllBinsFilled = 0;

			while (attempts < maxNumberOfAttempts) {
				if (distribute_X_Balls_in_N_bins_randomly_and_return_the_number_of_filled_bins(
						numberOfBalls, bins) == n) {
					numberOfTimesAllBinsFilled++;
				}
				++attempts;
			}

			float prob = numberOfTimesAllBinsFilled / maxNumberOfAttempts;
			System.out.println((prob > .5 ? "PASSED:" : "FAILED:") + numberOfBalls + ":" + prob);

		}
	}

	public static int distribute_X_Balls_in_N_bins_randomly_and_return_the_number_of_filled_bins(
			int x, boolean[] bins) {
		for (int inx = 0; inx < bins.length; ++inx) {
			bins[inx] = false;
		}

		int whichBin = 0;
		int total = 0;

		while (--x >= 0) {
			whichBin = rand.nextInt(bins.length);

			if (!bins[whichBin]) {
				bins[whichBin] = true;
				++total;

                                if (total == bins.length) {
					return total;
				}
			}
		}

		return total;

	}

}

[jackgolding]

Oh God that function