More efficiently calculate the value of a game that involves rerolling dice; see http://math.stackexchange.com/questions/1118263.

Question1118263.java

import java.math.BigInteger;
import java.util.Arrays;

public class Question1118263 {
	final static int [] values = {1,2,3,4,5,6};
	final static int nsides = values.length;
	final static int ndice = 20;

	static Rational [] v = new Rational [ndice];
	static int n;
	
	public static void main (String [] args) {
		Arrays.fill (v,Rational.ZERO);
		Rational p = Rational.ONE;
		for (n = 1;n < ndice;n++) {
			p = p.divide (BigInteger.valueOf (nsides));
			recurse (0,n,new int [nsides]);
			v [n] = v [n].multiply (p);
			System.out.println (n + " : " + v [n].subtract (v [n - 1]));
		}
	}
	
	static void recurse (int depth,int left,int [] counts) {
		if (depth == nsides - 1) {
			counts [depth] = left;
			int sum = 0;
			for (int i = 0;i < counts.length;i++)
				sum += counts [i] * values [i];
			int [] dice = new int [n];
			for (int i = 0,k = 0;i < counts.length;i++)
				for (int j = 0;j < counts [i];j++)
					dice [k++] = values [i];
			Rational max = Rational.ZERO;
			for (int reroll = 0;reroll < n;reroll++) {
				Rational newValue = v [reroll].add (sum);
				if (newValue.compareTo (max) > 0)
					max = newValue;
				sum -= dice [reroll];
			}
			BigInteger multinomial = BigInteger.ONE;
			int upper = n;
			for (int i = 0;i < counts.length;i++) {
				multinomial = multinomial.multiply (binomial (upper,counts [i]));
				upper -= counts [i];
			}
			v [n] = v [n].add (max.multiply (multinomial));
		}
		else
			for (counts [depth] = 0;counts [depth] <= left;counts [depth]++)
				recurse (depth + 1,left - counts [depth],counts);
	}
	
	static BigInteger [] [] binomials = new BigInteger [20] [20];
	
	static BigInteger binomial (int n,int k) {
		if (k > n)
			return BigInteger.ZERO;
		
		if (binomials [n] [k] == null) {
			binomials [n] [k] = BigInteger.ONE;
			for (int i = 0;i < k;i++)
				binomials [n] [k] = binomials [n] [k].multiply (BigInteger.valueOf (n - i)).divide (BigInteger.valueOf (i + 1));
		}
		return binomials [n] [k];
	}
}