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

(a) Find a recurrence relation for the number of moves required to solve the puzzle for n disks with this added restriction. Justify your answer

Answer 1

The recurrence relation for the number of moves required to solve the Tower of Hanoi puzzle with the added restriction is T(n) = 2T(n-1) + 1.

Step-by-step explanation:

The recurrence relation for the number of moves required to solve the Tower of Hanoi puzzle with the added restriction can be expressed as:

T(n) = 2T(n-1) + 1

Where T(n) represents the number of moves required to transfer n disks from peg 1 to peg 3.

This recurrence relation is justified by considering the following:

1. To transfer n disks from peg 1 to peg 3, we first need to transfer n-1 disks from peg 1 to peg 2. This requires T(n-1) moves.
2. Then, we need to move the largest disk from peg 1 to peg 3. This requires 1 move.
3. Finally, we need to transfer the remaining n-1 disks from peg 2 to peg 3. This again requires T(n-1) moves.

Therefore, the total number of moves is 2T(n-1) + 1