Tower of Hanoi

Definition

The Tower of Hanoi is a classic mathematical puzzle that involves moving a stack of discs from one peg to another, following a set of rules. It is often used as an example to demonstrate the power of recursive problem-solving techniques.

5 Must Know Facts For Your Next Test

1. The Tower of Hanoi puzzle consists of three pegs and a stack of discs of different sizes, with the largest disc at the bottom and the smallest at the top.
2. The objective is to move the entire stack from the starting peg to the destination peg, following the rules that only one disc can be moved at a time and each move involves taking the upper disc from one of the stacks and placing it on top of another stack or an empty peg.
3. The Tower of Hanoi problem can be solved recursively, where the solution for a stack of $n$ discs involves solving the problem for a stack of $n-1$ discs, moving the top $n-1$ discs to the intermediate peg, moving the largest disc to the destination peg, and then moving the $n-1$ discs from the intermediate peg to the destination peg.
4. The recursive solution to the Tower of Hanoi problem has a time complexity of $O(2^n)$, where $n$ is the number of discs, making it an exponential-time algorithm.
5. The Tower of Hanoi problem is often used as an example to illustrate the concept of recursion and the divide-and-conquer problem-solving strategy, as well as to analyze the efficiency of recursive algorithms.

Review Questions

• Explain the rules and objective of the Tower of Hanoi puzzle.
  • The Tower of Hanoi puzzle consists of three pegs and a stack of discs of different sizes, with the largest disc at the bottom and the smallest at the top. The objective is to move the entire stack from the starting peg to the destination peg, following the rules that only one disc can be moved at a time and each move involves taking the upper disc from one of the stacks and placing it on top of another stack or an empty peg.
• Describe the recursive approach to solving the Tower of Hanoi problem.
  • The Tower of Hanoi problem can be solved recursively, where the solution for a stack of $n$ discs involves solving the problem for a stack of $n-1$ discs. The steps are: (1) move the top $n-1$ discs to the intermediate peg, (2) move the largest disc to the destination peg, and (3) move the $n-1$ discs from the intermediate peg to the destination peg. This recursive approach breaks down the problem into smaller, similar subproblems, which can be solved efficiently using the divide-and-conquer strategy.
• Analyze the time complexity of the recursive solution to the Tower of Hanoi problem and explain why it is considered an exponential-time algorithm.
  • The recursive solution to the Tower of Hanoi problem has a time complexity of $O(2^n)$, where $n$ is the number of discs. This is because the number of moves required to solve the problem for a stack of $n$ discs is $2^n - 1$. The recursive nature of the solution, where the problem is broken down into smaller subproblems, leads to an exponential growth in the number of steps required as the number of discs increases. This makes the Tower of Hanoi problem an example of an exponential-time algorithm, which is considered computationally expensive and not scalable for large problem sizes.

Related terms

Recursion: A programming technique where a function calls itself to solve a problem by breaking it down into smaller, similar subproblems.

Algorithmic Efficiency: The measure of how well an algorithm performs in terms of time and space complexity, often analyzed using Big O notation.

Divide-and-Conquer: A problem-solving strategy that involves breaking a problem into smaller, more manageable subproblems, solving each subproblem, and then combining the solutions.