Today we will learn the Tower of Hanoi in C and also learn the tower of Hanoi program in C using recursion. So before starting, we will explain in detail about the Tower of Hanoi.

## What is Tower of Hanoi?

The Tower of Hanoi is the mathematical Puzzle. This puzzle was invented by the French mathematician **Edouard Lucas **in 1883.

There is a big dome situated in Banaras, that marks the middle of the world, contains a brass plate in which three diamond needles are fixed.

Each needle is cubit high and as thick as the body of a Bee.

When these needles were created God placed **sixty-four discs of pure gold** on one of this needle. The greatest disc resting on the brass plate, along with the others getting smaller and smaller up into the upper.

This Tower is also called the **Tower of Bramah Puzzle**.

Day and night unceasingly the priests move the disks from 1 diamond needle to another in accordance with the fixed and immutable laws of Bramah.

The priest on duty must not transfer more than 1 disk at a time and he should put this disk onto a needle so that there’s not any more smaller disk under it.

According to the legend when the last move of the puzzle will compete the world will vanish.

**The main Objective of the puzzle is to move the entire stack on the next rod.**

**Initial State:**

**Goal State:**

## Rules of Hanoi Puzzle:

*Only one disk may move at a time.*

*Each move consists of taking upper disk of one of the peg and put that disk on the other peg on top of the other disks.*

*None of the disk should place on top of smaller disk.*

## Applications of Tower of Hanoi:

There exists a version of the Job Known as Tower of London for neuropsychological analysis and therapy of executive purposes.

The Tower of Hanoi can be used as a Backup spinning strategy when performing computer information Backups where multiple Tapes and media are included.

As stated previously, the Tower of Hanoi is a favourite for instructing recursive algorithms to begin programming students. A pictorial version of the puzzle is programmed to the emacs editor, obtained by scanning M-x Hanoi.

The Tower of Hanoi can be utilized as a memory evaluation by neuropsychologists hoping to appraise amnesia.

**One Disk Solution:**

It contains only one move.

**Two Disk Solution:**

Note that we perform a solution to one disk tower two times one at the top row and second in the bottom row. Between rows, we move the bottom disk. Therefore we require **2*1+1=3 Moves.**

**Three Disk Solution:**

Note that we perform the solution for two disk tower two times once in the top row and once in the bottom row. Between rows, we moved to** **disk. Therefore we require **2(2*1+1)+1=7 Moves**.

**Nth Disk Analysis:**

Let **M(n)** denotes no if legal moves required to complete the tower of Hanoi puzzle which consist of n disks. Before the largest i.e nth disk can be moved to rightmost peg and **(n-1) **i.e remaining peg must move to the centre peg.

So this requires M(n-1) legal moves because these disks must be somewhere.

Thus we obtain relation of recursion i.e **M(n)=2M(n-1)+1**.

## Tower of Hanoi Solution:

Referring to the solution for a single disk **M(1)=1.**

The recursion relation **M(n)=2M(n-1)+1.**

which defines the solution **M(n)=2^n-1.**In the algorithms, this will be called exponential or **O(2^n)**.

But the difficulty us that they grow quickly out of hand.1 century is equivalent to **4.5+10^9 seconds**. So the age of the universe is **4 x 10^17 seconds**.

If the number of disks is odd then the first move should be transferred to the top of the** right peg.**

And if the number of disks is even then the first move should be transferred to the top of the **centre peg**.

So the puzzle is solved, The length of the shortest solution path to nth disk puzzle is** 2^n**.

**State Representation Using five disk Puzzle**:

Notation **(2 1 3 1 1)** indicates that

The smallest disk is on peg 2 (centre peg). The next larger disk is on peg 1 (left peg) and the next larger disk is on peg 3 (right peg). The next larger disk is on peg 1 and the next largest disk is on peg 1 which is largest.

## State Graph:

Each State is represented by a **labeled vertex** and legal moves are represented by** edges**.

**1 Disk:**

**2 Disk:**

**3 Disk:**

## 1. Tower of Hanoi Program in C

#include<stdio.h> int main () { int n; printf("Enter number of disks required: \n"); scanf ("%d", &n); TOH (n, 'A', 'B',' C'); return 0; } void TOH (int n, char src, char spare, char dest) { if (n==1) printf("Move from %c to %c \n", src, dest); else { TOH(n-1, src, dest, spare) ; TOH(1, src, spare, dest); TOH(n-1, spare, src, dest); } }

**Output:**

In the above example we have used 3 disks.

## 2. Tower of Hanoi in C Using Recursion

When the function calls itself is called recursion. In this program, we have declared the user-defined function called towers().

#include <stdio.h> void towers(int, char, char, char); int main() { int num; printf("Enter the number of disks : "); scanf("%d", &num); printf("The sequence of moves involved in the Tower of Hanoi are :\n"); towers(num, 'A', 'C', 'B'); return 0; } void towers(int num, char frompeg, char topeg, char auxpeg) { if (num == 1) { printf("\n Move disk 1 from peg %c to peg %c", frompeg, topeg); return; } towers(num - 1, frompeg, auxpeg, topeg); printf("\n Move disk %d from peg %c to peg %c", num, frompeg, topeg); towers(num - 1, auxpeg, topeg, frompeg); }

**Output:**

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