Question

In the Tower of Hanoi puzzle, suppose our goal is to transfer all n disks from peg 1 to peg 3 , but we cannot move a disk directly between pegs 1 and 3 . Each move of a disk must be a move involving peg 2 . As usual, we cannot place a disk on top of a smaller disk.

Find a recurrence relation for the number of moves required to solve the puzzle for n disks with this added restriction. Write out the steps.

Answer 1

The recurrence relation for the number of moves required to solve the Tower of Hanoi puzzle with the restriction of using peg 2 as an intermediate point is T(n) = 2 * T(n-1) + 1 for n > 1, with T(1) = 1 serving as the base case.

Step-by-step explanation:

The Tower of Hanoi puzzle is a classic problem in recursion and algorithm design. With the added restriction that in each move a disk can only be moved to or from the intermediate peg (peg 2), a different approach is needed compared to the standard Tower of Hanoi solution. To solve the puzzle for n disks with this constraint, we can derive a recurrence relation for the number of moves required.

1. First, we recognize that to move the largest disk (disk n), we must first move the top n-1 disks off of it.
2. To move the n-1 disks to peg 2, we follow the standard Tower of Hanoi steps for n-1 disks. This takes 2n-1 - 1 moves.
3. Once peg 2 is free, we can move the largest disk (disk n) to peg 3, which will require 1 move.
4. Finally, we must move the n-1 disks from peg 2 to peg 3, doubling the initial n-1 steps since each move must pass through peg 1 or 3, resulting in an additional 2(2n-1 - 1) moves.

Adding all these steps together gives us a total number of moves of T(n) = 2 * T(n-1) + 1 for n > 1, with T(1) = 1 as the base case, because when there is only one disk, we move it directly to peg 3 in one move.