Question

You will implement the simple Tower of Hanoi problem using Python. You can find the implementation in just about any book that talks about recursion. Here is the problem description: o There are n disks labled 1,2,3, ..., n and the three towers A, B and C. o No disk can be on top of a smaller disk at anytime. o All the disks are initially placed on tower A. o Only one disk can be moved at a time, and it must be the smallest disk on a tower. What you need to do: 1. Study and understand how recursion works for this problem. 2. Write a Python function and test it first. 3. Paste your Python code 4. Paste your run solution for n=2, n=3 and n=4.

Answer 1

In 1983 Edouard Lucas a French mathematician invented the Tower of Hanoi Problem.

The main constraints of the puzzle are:

1. Should move only one disk at a time
2. Should not place larger disk on the top of the smaller disk
3. Should move only top of the disk

Figure 1 shows an example of a outline of disks in the middle of a move from the first tower to the third tower. As per the rules, the disks on each tower are placed so that smaller disks are always on top of the larger disks.

solving this problem recursively

Suppose we are having a tower of five disks, originally on tower one.

Here is a high-level outline of how to move a tower from the starting tower, to the ending tower, using an intermediate tower:

1. Move a tower of height-1 to an intermediate tower, using the final tower.
2. Move the remaining disk to the final tower.
3. Move the tower of height-1 from the intermediate tower to the final tower using the original tower.

Here the Tower of Hanoi problem is a tower of one disk. In this case, we need move only a single disk to its final destination tower. A tower of one disk will be our base case

1. Create function TOH which takes the number of disk n and the names of the tower1, tower2 and tower3 as arguments.
2. Base case: When number of disks is n=1, int this case move disk from tower1 to tower3 and return.
3. Move n – 1 disks from tower1 to tower2 using the tower3 as the intermediate.
4. Move the one remaining disk on the tower1 to the tower3.
5. Move the n – 1 disks on the tower2 to the tower3 using the tower1 as the intermediate.

For n disks, total 2ⁿ – 1 moves are required.

Ex:

If n=2

Total moves from source to destination 2²-1= 3

If n=3

Total moves from source to destination 2³-1= 7

If n=4

Total moves from source to destination 2⁴-1= 15

#Python Implementation

def TOH(disk, Tower1, IntermediateTower2, Finaltower3):
if disk == 1:
print('Move disk 1 from Tower {} to Tower {}.'.format(Tower1, Finaltower3)) # base case
return

TOH(disk - 1, Tower1, Finaltower3, IntermediateTower2)  #here tower2 acts as the intermendiate tower
print('Move disk {} from Tower {} to Tower {}.'.format(disk, Tower1, Finaltower3))
TOH(disk - 1, IntermediateTower2, Tower1, Finaltower3)  # here tower1 acts as the intermediate tower


disk = int(input('Enter number of disks: '))
TOH(disk, 'A', 'B', 'C')
# here A is the Sourse tower i.e first tower1

# B is the Intermediate Tower i.e tower2

# C is the taget Tower i.e Finaltower3

#Output

>>>
===================== RESTART: E:/Desktop 7/HomeworkLib/TOH.py =====================
Enter number of disks: 2
Move disk 1 from Tower A to Tower B.
Move disk 2 from Tower A to Tower C.
Move disk 1 from Tower B to Tower C.
>>>
===================== RESTART: E:/Desktop 7/HomeworkLib/TOH.py =====================
Enter number of disks: 3
Move disk 1 from Tower A to Tower C.
Move disk 2 from Tower A to Tower B.
Move disk 1 from Tower C to Tower B.
Move disk 3 from Tower A to Tower C.
Move disk 1 from Tower B to Tower A.
Move disk 2 from Tower B to Tower C.
Move disk 1 from Tower A to Tower C.
>>>
===================== RESTART: E:/Desktop 7/HomeworkLib/TOH.py =====================
Enter number of disks: 4
Move disk 1 from Tower A to Tower B.
Move disk 2 from Tower A to Tower C.
Move disk 1 from Tower B to Tower C.
Move disk 3 from Tower A to Tower B.
Move disk 1 from Tower C to Tower A.
Move disk 2 from Tower C to Tower B.
Move disk 1 from Tower A to Tower B.
Move disk 4 from Tower A to Tower C.
Move disk 1 from Tower B to Tower C.
Move disk 2 from Tower B to Tower A.
Move disk 1 from Tower C to Tower A.
Move disk 3 from Tower B to Tower C.
Move disk 1 from Tower A to Tower B.
Move disk 2 from Tower A to Tower C.
Move disk 1 from Tower B to Tower C.
>>>