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# The Tower of Hanoi

The Tower of Hanoi is a classic problem in the world of programming. The problem setup consists of three rods/pegs and n disks.

The disks can be moved from one peg to another. The n disks are of different sizes.

Initially all the disks are stacked on the first tower. The disks are stacked in such a way that a disk is always over a disk bigger than itself.

## Tower of Hanoi Default Setup

Tower of Hanoi setup

## Historical Background

Fun fact: This game was invented by a French mathematician Édouard Lucas in the 19th century. It is associated with a legend of a Hindu temple where the puzzle was supposedly used to increase the mental discipline of young priests.

## Problem Statement

Move all the disks stacked on the first tower over to the last tower using a helper tower in the middle. While moving the disks, certain rules must be followed. These are:

- Only one disk can be moved.
- A larger disk cannot be placed on a smaller disk.

So you need to move all the disks from the first tower over to the last. You can only move one disk at a time and never place a smaller disk over a larger disk.

To do this you have an extra tower, it is known as helper/auxiliary tower.

Since you can only move one disk at a time, the disk you move will have to be at the top of its tower.

## Theoretical Solution to the Tower of Hanoi Problem

Let’s name the towers as A, B, C and the disks as 1, 2, 3.

Tower of Hanoi

We solve this question using simple recursion. To get the three disks over to the final tower you need to:

- Take the disk number 1 and 2 to tower B.
- Move disk number 3 to tower C.
- Take disk number 1 and 2 from B to C.

Of course, you can’t do it like this because of the constraints. However, we can use this to create a function that does it recursively.

## TOH 1 Starting state

In the function we write, we will print the movement of the disks.

## Implementing the Solution to Tower of Hanoi in Java

Let’s begin with understanding the two main parts of the code to solve the Tower of Hanoi problem.

**Base case:**The base case in our code is when we only have one disk. That is n=1.

if (n == 1) { System.out.println("Take disk 1 from rod " + from_rod + " to rod " + to_rod); return; }

**Recursive calls:**The recursive calls to solve tower of Hanoi are as follows:

towerOfHanoi(n-1, from_rod, helper_rod, to_rod); System.out.println("Take disk " + n + " from rod " + from_rod + " to rod " + to_rod); towerOfHanoi(n-1, helper_rod, to_rod, from_rod);

These are equivalent to:

- Move the top n-1 disks to the auxiliary tower.
- Move 1 disk from source rod to destination rod.
- Take the n-1 disks from auxiliary disk to the destination rod.

## Complete Java Implementation of Tower of Hanoi

package com.JournalDev; public class Main { static void towerOfHanoi(int n, char from_rod, char to_rod, char helper_rod) { if (n == 1) { System.out.println("Take disk 1 from rod " + from_rod + " to rod " + to_rod); return; } towerOfHanoi(n-1, from_rod, helper_rod, to_rod); System.out.println("Take disk " + n + " from rod " + from_rod + " to rod " + to_rod); towerOfHanoi(n-1, helper_rod, to_rod, from_rod); } public static void main(String args[]) { int n = 5; towerOfHanoi(n,'A','C', 'B'); } }

## Tower of Hanoi Output

The output for the code is:

Take disk 1 from rod A to rod C Take disk 2 from rod A to rod B Take disk 1 from rod C to rod B Take disk 3 from rod A to rod C Take disk 1 from rod B to rod A Take disk 2 from rod B to rod C Take disk 1 from rod A to rod C ...

We can understand the process using the following illustration. You can run the code for any number of disks.

## TOH for n=5

n=5

The formula to calculate the number of steps for n disks is:

## Conclusion

This is how you solve the Tower of Hanoi using recursion. Hope you had fun learning with us!

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