The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower,[1] 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:

1. Only one disk may be moved at a time.

2.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.

3. No disk may be placed on top of a smaller disk.

Origins:

The puzzle was first publicized in the West by the French mathematician Ã‰douard Lucas in 1883. There is a history about an Indian temple in Kashi Vihswanath which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end.[2] It is not clear whether Lucas invented this legend or was inspired by it.

Solution:

The following solution is a simple solution for the toy puzzle.

Alternate moves between the smallest piece and a non-smallest piece. When moving the smallest piece, always move it to the next position in the same direction (to the right if the starting number of pieces is even, to the left if the starting number of pieces is odd). If there is no tower position in the chosen direction, move the piece to the opposite end, but then continue to move in the correct direction. For example, if you started with three pieces, you would move the smallest piece to the opposite end, then continue in the left direction after that. When the turn is to move the non-smallest piece, there is only one legal move. Doing this will complete the puzzle using the fewest number of moves to do so.

Recursive solution

A key to solving this puzzle is to recognize that it can be solved by breaking the problem down into a collection of smaller problems and further breaking those problems down into even smaller problems until a solution is reached. The following procedure demonstrates this approach.

label the pegs A, B, Câ€”these labels may move at different steps

let n be the total number of discs

number the discs from 1 (smallest, topmost) to n (largest, bottommost)

To move n discs from peg A to peg C:

1. move n−1 discs from A to B. This leaves disc n alone on peg A

2.move disc n from A to C

3. move n−1 discs from B to C so they sit on disc n

The above is a recursive algorithm: to carry out steps 1 and 3, apply the same algorithm again for n−1. The entire procedure is a finite number of steps, since at some point the algorithm will be required for n = 1. This step, moving a single disc from peg A to peg B, is trivial. This approach can be given a rigorous mathematical formalism with the theory of dynamic programming, and is often used as an example of recursion when teaching programming.