Twisted Tower of Hanoi Problem

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The basic version of the Tower of Hanoi can be found here.
It is a twisted Tower of Hanoi problem. In which, all rules are the same with an addition of a rule:
You can not move any disk directly from the first rod to last rod i.e., If you want to move a disk from the first rod to the last rod then you have to move the first rod to the middle rod first and then to the last one.

Approach:

Base Case: If the number of disks is 1, then move it to the middle rod first and then move it to the last rod.
Recursive Case: In the recursive case, the following steps will produce the optimal solution:(All these moves follow the rules of the twisted Tower of Hanoi problem)
1. We will move the first n-1 disks to the last rod first.
2. Then move the largest disk to the middle rod.
3. Move the first n-1 disk from the last rod to the first rod.
4. Move the largest disk from the middle rod to the last rod.
5. Move all n-1 disks from the first rod to the last rod.

Below is the implementation of the above approach:

C++

// C++ implementation
#include <iostream>
using namespace std;
 
// Function to print the moves
void twistedTOH(int n, char first,
                char middle, char last)
{
    // Base case
    if (n == 1) {
 
        cout << "Move disk " << n
             << " from rod " << first
             << " to " << middle
             << " and then to "
             << last << endl;
 
        return;
    }
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
 
    // Move largest disk from first to middle
    cout << "Move disk " << n
         << " from rod " << first
         << " to " << middle << endl;
 
    // Move n-1 disks from last to first
    twistedTOH(n - 1, last, middle, first);
 
    // Move nth disk from middle to last
    cout << "Move disk " << n
         << " from rod " << middle
         << " to " << last << endl;
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
}
 
// Driver's Code
int main()
{
    // Number of disks
    int n = 2;
 
    // Rods are in order
    // first(A), middle(B), last(C)
    twistedTOH(n, 'A', 'B', 'C');
 
    return 0;
}

Java

// Java implementation of the approach 
import java.util.*;
 
class GFG
{
 
// Function to print the moves
static void twistedTOH(int n, char first,
                char middle, char last)
{
    // Base case
    if (n == 1)
    {
 
        System.out.println("Move disk " + n + " from rod " +
                                   first + " to " + middle + 
                                    " and then to " + last);
 
        return;
    }
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
 
    // Move largest disk from first to middle
    System.out.println("Move disk " + n + 
                       " from rod " + first + 
                       " to " + middle);
 
    // Move n-1 disks from last to first
    twistedTOH(n - 1, last, middle, first);
 
    // Move nth disk from middle to last
    System.out.println("Move disk " + n + 
                       " from rod " + middle + 
                       " to " + last);
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
}
 
// Driver Code
public static void main(String[] args)
{
    // Number of disks
    int n = 2;
 
    // Rods are in order
    // first(A), middle(B), last(C)
    twistedTOH(n, 'A', 'B', 'C');
}
}
 
// This code is contributed by PrinciRaj1992

Python3

# Python3 implementation of above approach
 
# Function to print the moves 
def twistedTOH(n, first, middle, last): 
     
    # Base case 
    if (n == 1): 
 
        print("Move disk", n, "from rod", first, 
              "to", middle, "and then to", last) 
 
        return
 
    # Move n-1 disks from first to last 
    twistedTOH(n - 1, first, middle, last) 
 
    # Move largest disk from first to middle 
    print("Move disk", n, "from rod",
                 first, "to", middle) 
 
    # Move n-1 disks from last to first 
    twistedTOH(n - 1, last, middle, first) 
 
    # Move nth disk from middle to last 
    print("Move disk", n, "from rod", 
                 middle, "to", last) 
 
    # Move n-1 disks from first to last 
    twistedTOH(n - 1, first, middle, last)
 
# Driver Code 
 
# Number of disks 
n = 2
 
# Rods are in order 
# first(A), middle(B), last(C) 
twistedTOH(n, 'A', 'B', 'C') 
 
# This code is contributed by
# divyamohan123

C#

// C# implementation of the approach 
using System;
     
class GFG
{
 
// Function to print the moves
static void twistedTOH(int n, char first,
                       char middle, char last)
{
    // Base case
    if (n == 1)
    {
        Console.WriteLine("Move disk " + n + " from rod " +
                                  first + " to " + middle + 
                                   " and then to " + last);
 
        return;
    }
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
 
    // Move largest disk from first to middle
    Console.WriteLine("Move disk " + n + 
                      " from rod " + first + 
                      " to " + middle);
 
    // Move n-1 disks from last to first
    twistedTOH(n - 1, last, middle, first);
 
    // Move nth disk from middle to last
    Console.WriteLine("Move disk " + n + 
                      " from rod " + middle + 
                      " to " + last);
 
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
}
 
// Driver Code
public static void Main(String[] args)
{
    // Number of disks
    int n = 2;
 
    // Rods are in order
    // first(A), middle(B), last(C)
    twistedTOH(n, 'A', 'B', 'C');
}
}
     
// This code is contributed by PrinciRaj1992

Javascript

<script>
// Javascript program 
// Function to print the moves
function twistedTOH(n, first, middle, last)
{
    // Base case
    if (n == 1) {
   
        document.write("Move disk " + n
             + " from rod " + first
             + " to " + middle
             + " and then to "
             + last + "<br>");
   
        return;
    }
   
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
   
    // Move largest disk from first to middle
    document.write("Move disk " + n
         + " from rod " + first
         + " to " + middle + "<br>");
   
    // Move n-1 disks from last to first
    twistedTOH(n - 1, last, middle, first);
   
    // Move nth disk from middle to last
    document.write("Move disk " + n
         + " from rod " + middle
         + " to " + last + "<br>");
   
    // Move n-1 disks from first to last
    twistedTOH(n - 1, first, middle, last);
}
 
// driver code
// Number of disks
var n = 2;
 
// Rods are in order
// first(A), middle(B), last(C)
twistedTOH(n, 'A', 'B', 'C');
 
// This code contributed by shivani
</script>

Output:

Move disk 1 from rod A to B and then to C
Move disk 2 from rod A to B
Move disk 1 from rod C to B and then to A
Move disk 2 from rod B to C
Move disk 1 from rod A to B and then to C

Recurrence formula:

T(n) = T(n-1) + 1 + T(n-1) + 1 + T(n-1)
     = 3 * T(n-1) + 2

where n is the number of disks.

By solving this recurrence, the Time Complexity will be O(3ⁿ).
Auxiliary Space: O(n).

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