Problem:
Problem B: Cleaning Robot

Dr. Asimov, a robotics researcher, loves to research, but hates houseworks and his house were really dirty. So, he has developed a cleaning robot.

As shown in the following figure, his house has 9 rooms, where each room is identified by an alphabet:

The robot he developed operates as follows:

If the battery runs down, the robot stops.

If not so, the robot chooses a direction from four cardinal points with the equal probability, and moves to the room in that direction. Then, the robot clean the room and consumes 1 point of the battery.

However, if there is no room in that direction, the robot does not move and remains the same room. In addition, there is a junk room in the house where the robot can not enter, and the robot also remains when it tries to enter the junk room. The robot also consumes 1 point of the battery when it remains the same place.

A battery charger for the robot is in a room. It would be convenient for Dr. Asimov if the robot stops at the battery room when its battery run down.

Your task is to write a program which computes the probability of the robot stopping at the battery room.

Input

The input consists of several datasets. Each dataset consists of:

n
s t b

n is an integer that indicates the initial battery point. s, t, b are alphabets respectively represent the room where the robot is initially, the battery room, and the junk room.

The input ends with a dataset which consists of single 0 for n. Your program should not output for this dataset.

Output

For each dataset, print the probability as a floating point number in a line. The answer may not have an error greater than 0.000001.

Constraints

Judge data includes at most 100 data sets.

n ≤ 15

s, t, b are distinct.

Sample Input

1
E A C
1
E B C
2
E A B
0

Output for the Sample Input

0.00000000
0.25000000
0.06250000

Solution in JAVA (judge verdict: Accepted)


import java.io.IOException;
import java.util.Scanner;

public class Main {

	public static void main(String[] args) throws IOException {

		new Main().run();
	}

	private void run() throws IOException {
		Scanner scanner = new Scanner(System.in);
		while (true) {
			int n = scanner.nextInt();
			if (n == 0)
				break;
			int[] xs = new int[3];
			int[] ys = new int[3];
			for (int i = 0; i < 3; i++) {
				int a = scanner.next().charAt(0) - 'A';
				xs[i] = a % 3;
				ys[i] = a / 3;
			}
			double[][] cur = new double[3][3];
			cur[xs[0]][ys[0]] = 1;
			for (int i = 0; i < n; i++) {
				double[][] nxt = new double[3][3];
				for (int x = 0; x < 3; x++)
					for (int y = 0; y < 3; y++)
						for (int[] m : move) {
							int nx = x + m[0];
							int ny = y + m[1];
							if (nx == xs[2] && ny == ys[2] || nx < 0 || 2 < nx
									|| ny < 0 || 2 < ny)
								nxt[x][y] += cur[x][y] / 4;
							else
								nxt[nx][ny] += cur[x][y] / 4;
						}
				cur = nxt;
			}
			System.out.printf("%.8f\n", cur[xs[1]][ys[1]]);
		}
	}

	int[][] move = { { -1, 0 }, { 0, -1 }, { 0, 1 }, { 1, 0 } };

}