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The
Never Ending Towers of Hanoi
The Problem
In 1883, Edward Lucas invented, or perhaps reinvented, one of the most
popular puzzles of all times - the Tower of Hanoi, as he called it - which
is still used today in many computer science textbooks to demonstrate how
to write a recursive algorithm or program. First of all, we will make a
list of the rules of the puzzle:
· There are three pegs: A, B and C.
· There are n disks. The number n is constant while working
the puzzle.
· All disks are different in size.
· The disks are initially stacked on peg A so that they increase
in size from the top to the bottom.
· The goal of the puzzle is to transfer the entire tower from
the A peg to the peg C.
· One disk at a time can be moved from the top of a stack either
to an empty peg or to a peg with a larger disk than itself on the top of
its stack.
Your job will be to write a program which will show a copy of the puzzle
on the screen step by step, as you move the disks around. This program
has to solve the problem in an efficient way.
TIP: It is well known and rather easy to prove that the minimum number
of moves needed to complete the puzzle with n disks is 2n -1.
The Input
The input file will consist of a series of lines.
Each line will contain two integers n, m. n, lying within the range [1,250],
will denote the number of disks and m, belonging to [0,2n-1], will be
the number of the last move, you may assume that m will also be less than
216, and you may also assume that a good algorithm will always have enough
time. The file will end at a line formed by two zeros.
The Output
The output will consist again of a series of lines, formatted as show below.
NOTES: There are 3 spaces between de ‘=>’ and the first number
printed. If there isn't any number, there should be no spaces.
All the disks in a single peg are printed in a single line.
Print a blank line after every problem.
Sample
Input
64 2
8 45
0 0
Sample
Output
Problem #1
A=> 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
B=>
C=>
A=> 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2
B=> 1
C=>
A=> 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3
B=> 1
C=> 2
Problem #2
A=> 8 7 6 5 4 3 2 1
B=>
C=>
A=> 8 7 6 5 4 3 2
B=> 1
C=>
A=> 8 7 6 5 4 3
B=> 1
C=> 2
A=> 8 7 6 5 4 3
B=>
C=> 2 1
A=> 8 7 6 5 4
B=> 3
C=> 2 1
A=> 8 7 6 5 4 1
B=> 3
C=> 2
A=> 8 7 6 5 4 1
B=> 3 2
C=>
A=> 8 7 6 5 4
B=> 3 2 1
C=>
A=> 8 7 6 5
B=> 3 2 1
C=> 4
A=> 8 7 6 5
B=> 3 2
C=> 4 1
A=> 8 7 6 5 2
B=> 3
C=> 4 1
A=> 8 7 6 5 2 1
B=> 3
C=> 4
A=> 8 7 6 5 2 1
B=>
C=> 4 3
A=> 8 7 6 5 2
B=> 1
C=> 4 3
A=> 8 7 6 5
B=> 1
C=> 4 3 2
A=> 8 7 6 5
B=>
C=> 4 3 2 1
A=> 8 7 6
B=> 5
C=> 4 3 2 1
A=> 8 7 6 1
B=> 5
C=> 4 3 2
A=> 8 7 6 1
B=> 5 2
C=> 4 3
A=> 8 7 6
B=> 5 2 1
C=> 4 3
A=> 8 7 6 3
B=> 5 2 1
C=> 4
A=> 8 7 6 3
B=> 5 2
C=> 4 1
A=> 8 7 6 3 2
B=> 5
C=> 4 1
A=> 8 7 6 3 2 1
B=> 5
C=> 4
A=> 8 7 6 3 2 1
B=> 5 4
C=>
A=> 8 7 6 3 2
B=> 5 4 1
C=>
A=> 8 7 6 3
B=> 5 4 1
C=> 2
A=> 8 7 6 3
B=> 5 4
C=> 2 1
A=> 8 7 6
B=> 5 4 3
C=> 2 1
A=> 8 7 6 1
B=> 5 4 3
C=> 2
A=> 8 7 6 1
B=> 5 4 3 2
C=>
A=> 8 7 6
B=> 5 4 3 2 1
C=>
A=> 8 7
B=> 5 4 3 2 1
C=> 6
A=> 8 7
B=> 5 4 3 2
C=> 6 1
A=> 8 7 2
B=> 5 4 3
C=> 6 1
A=> 8 7 2 1
B=> 5 4 3
C=> 6
A=> 8 7 2 1
B=> 5 4
C=> 6 3
A=> 8 7 2
B=> 5 4 1
C=> 6 3
A=> 8 7
B=> 5 4 1
C=> 6 3 2
A=> 8 7
B=> 5 4
C=> 6 3 2 1
A=> 8 7 4
B=> 5
C=> 6 3 2 1
A=> 8 7 4 1
B=> 5
C=> 6 3 2
A=> 8 7 4 1
B=> 5 2
C=> 6 3
A=> 8 7 4
B=> 5 2 1
C=> 6 3
A=> 8 7 4 3
B=> 5 2 1
C=> 6
A=> 8 7 4 3
B=> 5 2
C=> 6 1