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 Ruby (judge verdict: Accepted)

while (n = gets.chomp.to_i) != 0
  s,t,b = gets.chomp.split

  rooms = {
    [0,0] => {name: "A", value: 0.0}, 
    [0,1] => {name: "B", value: 0.0}, 
    [0,2] => {name: "C", value: 0.0}, 
    [1,0] => {name: "D", value: 0.0}, 
    [1,1] => {name: "E", value: 0.0}, 
    [1,2] => {name: "F", value: 0.0}, 
    [2,0] => {name: "G", value: 0.0}, 
    [2,1] => {name: "H", value: 0.0}, 
    [2,2] => {name: "I", value: 0.0}
  } 

  rooms.each_value{|room| room[:value] = 1 if room[:name] == s}
  rooms.delete_if{|k, v| v[:name] == b}

  n.times do |n|
    tmp = {
      [0,0] => 0.0, [0,1] => 0.0, [0,2] => 0.0,
      [1,0] => 0.0, [1,1] => 0.0, [1,2] => 0.0,
      [2,0] => 0.0, [2,1] => 0.0, [2,2] => 0.0
    }
    rooms.each do |k, v|
      if v[:value] != 0.0
        row = k[0]
        column = k[1]
        p = v[:value] / 4.0
        #up
        if rooms.has_key?([row-1, column])
          rooms[k][:value] -= p
          tmp[[row-1, column]] += p
        else
          tmp[[row, column]] += p
        end
        #down
        if rooms.has_key?([row+1, column])
          rooms[k][:value] -= p
          tmp[[row+1, column]] += p
        else
          tmp[[row, column]] += p
        end
        #left
        if rooms.has_key?([row, column-1])
          rooms[k][:value] -= p
          tmp[[row, column-1]] += p
        else
          tmp[[row, column]] += p
        end
        #right
        if rooms.has_key?([row, column+1])
          rooms[k][:value] -= p
          tmp[[row, column+1]] += p
        else
          tmp[[row, column]] += p
        end
      end
    end
    rooms.each_key do |k|
      rooms[k][:value] = tmp[k]
    end
  end
  rooms.each_value do |v|
    puts v[:value] if v[:name] == t
  end
end