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Towers of Hanoi

Towers of Hanoi is a popular mathematical puzzle invented in 1883 by French mathematician Eduoardo Lucas. It is also a popular example in coding world because it is a typical example where a recursive solution is much easier than iterative solution.

In the puzzle, there are 3 rods and n discs of different sizes. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

Gif file from wikipedia
Aim of the game is to transfer these  n discs to the 3rd rod, using 2nd rod as temporary location, but keeping in mind that
  • only one disc can be moved at a time.
  • a larger disc can not be placed above a smaller disc.

How do we solve this puzzle?

If some how we move n-1 discs to 2nd rod in correct order, we only have largest disc in 1st rod. And it can be moved to 3rd rod.

So now we need to find a method of moving n-1 discs from 1st rod to 2nd rod in correct order. Again we can use 3rd rod for intermediate storage. Why don't we move n-2 rods from 1st to 3rd rod and then we only have to move n-1th disc from 1st to 2nd rod.

So this is recursion. And we can write our algorithm as
  • recursively move n-1 discs from source rod to extra rod
  • move nth disc from source rod to target rod
  • recursively move n-1 discs from extra rod to target rod
And here is C code for the function.

void move_discs(int numdisks,char from,char to, char middle)
{
   if(numdisks>0){
    move_discs(numdisks-1,from,middle,to);
    printf("Move disk %d from rod %c to rod %c\n",numdisks,from,to);
    move_discs(numdisks-1,middle,to,from);
  }
}

move_discs is called recursively for n-1 discs for moving from from rod to middle rod. And then nth rod is moved - which is represented by printf function.

next move_discs is called recursively for n-1 discs for moving from middle rod to to rod.

The base case is when n is 0, where recursion is not called.


For 3 discs, the output appears like this

aa@dell:~/dsPrograms$ ./a.out
Enter the number of discs3
Move disk 1 from rod A to rod C
Move disk 2 from rod A to rod B
Move disk 1 from rod C to rod B
Move disk 3 from rod A to rod C
Move disk 1 from rod B to rod A
Move disk 2 from rod B to rod C
Move disk 1 from rod A to rod C

And here we have driver program.


#include<stdio.h>
void move_discs(int numdisks,char from,char to, char middle)
{
   if(numdisks>0){
    move_discs(numdisks-1,from,middle,to);
    printf("Move disk %d from rod %c to rod %c\n",numdisks,from,to);
    move_discs(numdisks-1,middle,to,from);
  }
}
int main()
{
   int n;
   printf("Enter the number of discs");
   scanf("%d",&n);
   move_discs(n,'A','C','B');
   return 0;
}


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