I'm trying to understand how to prove that a recursive function works using mathematical induction. For example, with the Towers of Hanoi problem, I know that I need to: 1) Prove the code works for the base case of n=1 (i.e., moving one disk), 2) Assume that it works correctly for n disks, and 3) Prove it works for n+1 disks. However, I'm unclear on how to actually prove that the function works for n+1 disks. Any guidance would be appreciated!
4 Answers
Absolutely, you can use induction to reason about recursive functions like the Towers of Hanoi! The crucial part is showing that your algorithm works for the additional case after assuming it works for n disks. Essentially, you have three steps: 1) Move the n disks from the starting peg to a temporary peg (using your assumption), 2) Move the (n+1)th disk directly to the target peg, and 3) Finally, move the n disks from the temporary peg to the target peg. This structure keeps your moves valid! 👍
Great breakdown! It really shows how recursion chains together the smaller problems.
You can definitely prove that a recursive function works by using induction! The basic structure for Towers of Hanoi will follow this flow: first move the n disks to a spare peg, move the largest disk to the destination, then move the n disks over to complete the stack. Use the assumption that it works for n disks to handle those transitions, and your proof will hold!
Spot on! It’s a neat way to show the transitions between steps are valid.
It’s all about that inductive reasoning! Keep up the good work!
Yes, you can prove it using standard induction! Just do your induction proof by first showing it’s true for n=1. Then, for n+1, break it into manageable steps just like you mentioned. Validate that each part of your code does its job correctly in line with your assumptions. Simple enough, but really works! Math proofs can be fun!
Agreed! The joy of proving something works step by step is pretty satisfying.
You're on the right track! When proving it for n+1 disks, show that: 1) Your code successfully moves n disks from the first peg to the second peg, 2) It moves the (n+1)th disk to the third peg, and 3) Finally, it transfers the n disks from the second peg to the third peg. You can usually just rename your pegs to simplify the proof of the individual steps. It's all about connecting those parts logically!
So true! Each step is like a mini-proof that builds up to the solution.
Exactly! It’s like solving a puzzle step by step; each move relies on the previous one being correct.