Tower of Hanoi Problem in DSA Python - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Tower of Hanoi moves for 2 disks

What is the output of the following Tower of Hanoi function call for 2 disks?

DSA Python

def tower_of_hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk 1 from {source} to {target}")
        return
    tower_of_hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    tower_of_hanoi(n-1, auxiliary, target, source)

tower_of_hanoi(2, 'A', 'C', 'B')

A

Move disk 1 from A to B
Move disk 2 from A to C
Move disk 1 from B to C

B

Move disk 1 from A to C
Move disk 2 from A to B
Move disk 1 from C to B

C

Move disk 1 from B to C
Move disk 2 from A to C
Move disk 1 from A to B

D

Move disk 1 from C to B
Move disk 2 from B to A
Move disk 1 from C to A

💡 Hint

Remember the recursive steps: move n-1 disks to auxiliary, move nth disk to target, then move n-1 disks from auxiliary to target.

For 2 disks, the moves are: move disk 1 from source to auxiliary, move disk 2 from source to target, then move disk 1 from auxiliary to target.

🧠 Conceptual

intermediate

Minimum moves required for Tower of Hanoi

What is the minimum number of moves required to solve the Tower of Hanoi problem with 4 disks?

A 15

B 31

C 16

D 8

💡 Hint

The minimum moves follow the formula 2^n - 1, where n is the number of disks.

For 4 disks, minimum moves = 2^4 - 1 = 16 - 1 = 15.

❓ Predict Output

advanced

Output of Tower of Hanoi moves for 3 disks

What is the output of the following Tower of Hanoi function call for 3 disks?

DSA Python

def tower_of_hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk 1 from {source} to {target}")
        return
    tower_of_hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    tower_of_hanoi(n-1, auxiliary, target, source)

tower_of_hanoi(3, 'A', 'C', 'B')

A

Move disk 1 from A to B
Move disk 2 from A to C
Move disk 1 from B to C
Move disk 3 from A to C
Move disk 1 from C to B
Move disk 2 from C to A
Move disk 1 from B to A

B

Move disk 1 from A to B
Move disk 2 from A to C
Move disk 1 from B to C
Move disk 3 from A to B
Move disk 1 from C to A
Move disk 2 from C to B
Move disk 1 from A to B

C

Move disk 1 from A to C
Move disk 2 from A to B
Move disk 1 from C to B
Move disk 3 from A to C
Move disk 1 from B to A
Move disk 2 from B to C
Move disk 1 from A to C

D

Move disk 1 from B to C
Move disk 2 from A to B
Move disk 1 from C to A
Move disk 3 from A to C
Move disk 1 from A to B
Move disk 2 from B to C
Move disk 1 from B to C

💡 Hint

Trace the recursive calls carefully for 3 disks.

The correct sequence follows the recursive pattern moving n-1 disks to auxiliary, moving nth disk to target, then moving n-1 disks from auxiliary to target.

🔧 Debug

advanced

Identify the error in Tower of Hanoi code

What error will the following Tower of Hanoi code produce when called with n=2?

DSA Python

def tower_of_hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk 1 from {source} to {target}")
        return
    tower_of_hanoi(n-1, source, target, auxiliary)
    print(f"Move disk {n} from {source} to {target}")
    tower_of_hanoi(n-1, auxiliary, target, source)

tower_of_hanoi(2, 'A', 'C', 'B')

A SyntaxError due to missing colon

B Moves disks correctly with no error

C TypeError due to wrong argument types

D RecursionError: maximum recursion depth exceeded

💡 Hint

Check the order of parameters in recursive calls carefully.

The recursive call tower_of_hanoi(n-1, source, target, auxiliary) is incorrect; it should be tower_of_hanoi(n-1, source, auxiliary, target). This causes infinite recursion.

🧠 Conceptual

expert

Time complexity of Tower of Hanoi algorithm

What is the time complexity of the Tower of Hanoi recursive algorithm with n disks?

A O(n)

B O(2^n)

C O(n^2)

D O(n!)

💡 Hint

Each move doubles the previous moves plus one more move.

The Tower of Hanoi requires 2^n - 1 moves, so time complexity is exponential O(2^n).

Practice

1. What is the minimum number of moves required to solve the Tower of Hanoi problem with 3 disks?

easy

A. 5

B. 7

C. 9

D. 6

Solution

1. Step 1: Understand the formula for minimum moves

   The minimum moves required for n disks is 2^n - 1.

2. Step 2: Calculate for 3 disks

   For 3 disks, minimum moves = 2^3 - 1 = 8 - 1 = 7.

3. Final Answer:

   7 -> Option B

4. Quick Check:

   2^3 - 1 = 7 [OK]

Hint: Use formula 2^n - 1 for minimum moves [OK]

Common Mistakes:

• Using n instead of 2^n - 1
• Calculating 2*n - 1 instead of 2^n - 1
• Confusing number of disks with moves

2. Which of the following is the correct base case condition in a recursive Tower of Hanoi function in Python?

easy

A. if n == 0: return

B. if n < 0: return

C. if n > 1: call recursion

D. if n == 1: move disk

Solution

1. Step 1: Identify base case for recursion

   The base case stops recursion when only one disk is left, i.e., n == 1.

2. Step 2: Understand base case action

   When n == 1, the disk is moved directly from source to target.

3. Final Answer:

   if n == 1: move disk -> Option D

4. Quick Check:

   Base case is n == 1 [OK]

Hint: Base case is when only one disk remains (n == 1) [OK]

Common Mistakes:

• Using n == 0 as base case incorrectly
• Not defining a base case causing infinite recursion
• Using invalid conditions like n < 0

3. What will be the output of the following Python code snippet for Tower of Hanoi with 2 disks?

def hanoi(n, source, target, auxiliary):
    if n == 0:
        return
    hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    hanoi(n-1, auxiliary, target, source)

hanoi(2, 'A', 'C', 'B')

medium

A. Move disk 1 from A to B
Move disk 2 from A to C
Move disk 1 from B to C

B. Move disk 2 from A to C
Move disk 1 from A to B
Move disk 1 from B to C

C. Move disk 1 from A to C
Move disk 2 from A to B
Move disk 1 from C to B

D. Move disk 1 from B to C
Move disk 2 from A to C
Move disk 1 from A to B

Solution

1. Step 1: Trace recursive calls for n=2

   First, move disk 1 from A to B, then disk 2 from A to C, then disk 1 from B to C.

2. Step 2: Verify print statements order

   Prints match the moves: disk 1 A->B, disk 2 A->C, disk 1 B->C.

3. Final Answer:

   Move disk 1 from A to B Move disk 2 from A to C Move disk 1 from B to C -> Option A

4. Quick Check:

   Correct move sequence for 2 disks [OK]

Hint: Follow recursive calls step-by-step [OK]

Common Mistakes:

• Mixing source and auxiliary rods
• Printing moves in wrong order
• Confusing disk numbers

4. Identify the error in this Tower of Hanoi recursive function and select the correct fix:

def hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk {n} from {source} to {target}")
    else:
        hanoi(n-1, source, target, auxiliary)
        print(f"Move disk {n} from {source} to {target}")
        hanoi(n-1, auxiliary, target, source)

medium

A. Change first recursive call to hanoi(n-1, source, auxiliary, target)

B. Change base case to if n == 0

C. Swap print statements order

D. Remove the else block

Solution

1. Step 1: Analyze recursive calls

   The first recursive call incorrectly moves disks from source to target instead of auxiliary.

2. Step 2: Correct the first recursive call

   It should be hanoi(n-1, source, auxiliary, target) to move disks to auxiliary rod first.

3. Final Answer:

   Change first recursive call to hanoi(n-1, source, auxiliary, target) -> Option A

4. Quick Check:

   Correct recursive call order fixes the error [OK]

Hint: First recursive call moves disks to auxiliary rod [OK]

Common Mistakes:

• Calling recursion with wrong target rod
• Changing base case incorrectly
• Swapping print statements unnecessarily

5. You want to solve Tower of Hanoi for 4 disks but only have 3 rods. What is the minimum number of moves required, and why can't it be less than that?

hard

A. 14 moves, because the largest disk can be moved directly

B. 16 moves, because one extra move is needed for 4 disks

C. 15 moves, because 2^4 - 1 is the minimum for 3 rods

D. 7 moves, because only half the disks need moving

Solution

1. Step 1: Apply minimum moves formula for 3 rods

   Minimum moves for n disks with 3 rods is 2^n - 1, so for 4 disks: 2^4 - 1 = 15.

2. Step 2: Understand why moves can't be less

   With 3 rods, no shortcut exists; all disks must be moved following rules, so 15 is minimum.

3. Final Answer:

   15 moves, because 2^4 - 1 is the minimum for 3 rods -> Option C

4. Quick Check:

   Minimum moves = 2^n - 1 for 3 rods [OK]

Hint: Use 2^n - 1 formula for 3 rods minimum moves [OK]

Common Mistakes:

• Assuming fewer moves possible with 3 rods
• Adding extra moves unnecessarily
• Confusing number of rods with moves