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.util.*;
import java.math.*;
import java.awt.geom.*;
import java.io.*;
      
      
public class Main {
	static int[] vx = new int[]{0,1,0,-1};
	static int[] vy = new int[]{1,0,-1,0};
	static int[] Y  = new int[]{0,0,0,1,1,1,2,2,2};
	static int[] X  = new int[]{0,1,2,0,1,2,0,1,2};
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		while(true) {
			int n = sc.nextInt();
			if(n == 0) break;
			char s = sc.next().charAt(0);
			int S = s - 'A';
			char t = sc.next().charAt(0);
			int T = t - 'A';
			char b = sc.next().charAt(0);
			int B = b - 'A';
			int[][][] dp = new int[n+1][3][3];
			dp[0][Y[S]][X[S]] = 1;
			for(int i = 0; i < n; i++) {
				for(int j = 0; j < 3; j++) {
					for(int k = 0; k < 3; k++) {
						for(int l = 0; l < 4; l++) {
							int ty = j + vy[l];
							int tx = k + vx[l];
							if(ty < 0 || tx < 0 || tx >= 3 || ty >= 3 || (ty == Y[B] && tx == X[B])) {
								ty = j;
								tx = k;
							}
							dp[i+1][ty][tx] += dp[i][j][k];
						}
					}
				}
			}
			int sum = 0;
			for(int i = 0; i < 3; i++) {
				for(int j = 0; j < 3; j++) {
					sum += dp[n][i][j];
				}
			}
			System.out.printf("%.25f\n",dp[n][Y[T]][X[T]] / (double)sum);
			
		}
	}
 }