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 C (judge verdict: Time Limit Exceeded)

#include <stdio.h>

int patterns = 0;
int goals = 0;
/* typedef enum { */
/* 	A = 0, */
/* 	B, */
/* 	C, */
/* 	D, */
/* 	E, */
/* 	F, */
/* 	G, */
/* 	H, */
/* 	I, */
/* } ROOMS; */
void cleaning(char lroom, char groom, char broom, int cnt);
void subcleaning(char lroom, char groom, char broom, int cnt, char next);

void subcleaning(char lroom, char groom, char broom, int cnt, char next) {
//	printf("%c ", next);
	
	if (next == broom || next == 'N') {
		cleaning(lroom, groom, broom, --cnt);
	} else {
		cleaning(next, groom, broom, --cnt);
	}
}

void cleaning(char lroom, char groom, char broom, int cnt) {
//	++patterns;
	if (0 == cnt) {
		++patterns;
		if (lroom == groom) {
			++goals;
		}
		return;
	}
	
	switch(lroom) {
	case 'A':
		subcleaning(lroom, groom, broom, cnt, 'B');
		subcleaning(lroom, groom, broom, cnt, 'D');
		subcleaning(lroom, groom, broom, cnt, 'N');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'B':
		subcleaning(lroom, groom, broom, cnt, 'A');
		subcleaning(lroom, groom, broom, cnt, 'E');
		subcleaning(lroom, groom, broom, cnt, 'C');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'C':
		subcleaning(lroom, groom, broom, cnt, 'B');
		subcleaning(lroom, groom, broom, cnt, 'F');
		subcleaning(lroom, groom, broom, cnt, 'N');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'D':
		subcleaning(lroom, groom, broom, cnt, 'A');
		subcleaning(lroom, groom, broom, cnt, 'E');
		subcleaning(lroom, groom, broom, cnt, 'G');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'E':
		subcleaning(lroom, groom, broom, cnt, 'B');
		subcleaning(lroom, groom, broom, cnt, 'D');
		subcleaning(lroom, groom, broom, cnt, 'F');
		subcleaning(lroom, groom, broom, cnt, 'H');
		break;
	case 'F':
		subcleaning(lroom, groom, broom, cnt, 'C');
		subcleaning(lroom, groom, broom, cnt, 'E');
		subcleaning(lroom, groom, broom, cnt, 'I');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'G':
		subcleaning(lroom, groom, broom, cnt, 'D');
		subcleaning(lroom, groom, broom, cnt, 'H');
		subcleaning(lroom, groom, broom, cnt, 'N');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'H':
		subcleaning(lroom, groom, broom, cnt, 'G');
		subcleaning(lroom, groom, broom, cnt, 'E');
		subcleaning(lroom, groom, broom, cnt, 'I');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	case 'I':
		subcleaning(lroom, groom, broom, cnt, 'F');
		subcleaning(lroom, groom, broom, cnt, 'I');
		subcleaning(lroom, groom, broom, cnt, 'N');
		subcleaning(lroom, groom, broom, cnt, 'N');
		break;
	}
	return;
}

int main(void) {
	int n;
	char s, t, b; 
	while(1) {
		scanf("%d", &n);
		if (n == 0) {
			break;
		}
		scanf(" %c %c %c", &s, &t, &b);
//		printf("%c %c %c\n", s, t, b);
		patterns = 0;
		goals = 0;
		cleaning(s, t, b, n);
		printf("goals = %d\n", goals);
		printf("patterns = %d\n", patterns);
		printf("%lf\n", (double)goals/patterns);
	}
	return 0;
}