Tower of Hanoi Solver

The classic mathematical puzzle that demonstrates recursive algorithms. Visualize the solution step by step!

How to Play: The objective is to move all disks from Tower A to Tower C, using Tower B as auxiliary. Only one disk can be moved at a time, and a larger disk cannot be placed on top of a smaller disk.

Auto-Solve Mode
Step-by-Step Mode
0
Moves
5
Disks
31
Minimum Moves
00:00
Time
Minimum moves: 31
Animation Speed:
Puzzle Solved!

You've solved the Tower of Hanoi with 5 disks in 0 moves!

The optimal solution requires 31 moves (2^n - 1).

Move History

Moves: 0

About Tower of Hanoi

The Tower of Hanoi is a mathematical puzzle invented by the French mathematician Édouard Lucas in 1883. 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.

Mathematical Significance: The Tower of Hanoi is often used in computer science to teach recursive algorithms. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2^n - 1, where n is the number of disks.

Recursive Algorithm

function moveTower(n, source, destination, auxiliary) {

  if (n === 1) {

    // Move disk 1 from source to destination

    moveDisk(source, destination);

  } else {

    // Move n-1 disks from source to auxiliary

    moveTower(n-1, source, auxiliary, destination);

    // Move the nth disk from source to destination

    moveDisk(source, destination);

    // Move n-1 disks from auxiliary to destination

    moveTower(n-1, auxiliary, destination, source);

  }

}

This recursive algorithm elegantly solves the Tower of Hanoi puzzle. For n disks, the algorithm makes 2^n - 1 moves, which is provably optimal.

Mathematical Properties

Number of Disks (n) Minimum Moves (2^n - 1) Time Complexity Recursive Calls Difficulty Level
3 7 O(2^n) 15 Easy
4 15 O(2^n) 31 Easy
5 31 O(2^n) 63 Medium
6 63 O(2^n) 127 Medium
7 127 O(2^n) 255 Hard
8 255 O(2^n) 511 Hard
9 511 O(2^n) 1023 Expert

Game Variations

Classic Tower of Hanoi: The standard puzzle with three towers and n disks. The goal is to move all disks from the first tower to the third tower.

1

Reve's Puzzle (Four Pegs): A variation with four pegs instead of three. This is also known as the "Tower of Hanoi with four pegs" and has a more complex optimal solution.

2

Cyclic Hanoi: A variation where the disks can only be moved in one direction (clockwise or counterclockwise) around the three pegs.

3

Tower of Hanoi with Forbidden Moves: A variation where certain moves between pegs are not allowed, adding additional constraints to the puzzle.

Frequently Asked Questions

For n disks, the minimum number of moves required is 2^n - 1. This has been proven to be optimal. So for 3 disks, it's 7 moves; for 4 disks, it's 15 moves; for 5 disks, it's 31 moves, and so on.

Tower of Hanoi is a classic example of a recursive algorithm. It demonstrates how a complex problem can be broken down into smaller, similar subproblems. This is a fundamental concept in algorithm design and is used in many areas of computer science including divide-and-conquer algorithms, recursion, and complexity analysis.

Yes, there are several patterns. One well-known pattern is that for an odd number of disks, the first move is from the source to the destination peg. For an even number of disks, the first move is from the source to the auxiliary peg. Additionally, the smallest disk moves every other move, alternating between two specific pegs.

Yes, Tower of Hanoi can be solved iteratively using a simple algorithm: For an even number of disks, make legal moves between pegs A and B, then A and C, then B and C, repeating until solved. For an odd number of disks, make legal moves between pegs A and C, then A and B, then C and B, repeating until solved. This iterative solution produces exactly the same moves as the recursive solution.

According to the legend, there is a temple in India where priests are moving 64 golden disks according to the rules of the Tower of Hanoi. When they finish moving all 64 disks, the world will end. Since it takes 2^64 - 1 moves (approximately 18.4 quintillion moves), even at one move per second, it would take about 584 billion years to complete - much longer than the current age of the universe!