Category Archives: Combinatorics
The Tower of Hanoi (also called the Tower of Brahma or Lucas’ Tower, and sometimes pluralised) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. 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.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
- Only one disk may be moved at a time.
- Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
- No disk may be placed on top of a smaller disk.
With three disks, the puzzle can be solved in seven moves.
for disks and towers, the number of steps required to transfer the disks from tower to another is .
Proof of our lemma:
Now we transfer the th disc to peg 2 . One move will be required for this.
Finally we will transfer discs which are on peg to peg . This will require another moves. Thus the transfer will be done in moves which will be equal to movesTherefore we have formed a recurrence relation that
Now we will obtain a general formula for
We will multiply both sides of the above relations by
Solving this gives us
Since ( one move will be required to move )
THIS FINALLY GIVES
1. wikepedia ( for the introduction) : http://en.wikipedia.org/wiki/Tower_of_Hanoi
Find the number of integers in the set such that is a multiple of