So, in total, you will need m + 1 + m = (2m + 1) steps to solve the puzzle. Next, we will move back all the 3 disks from peg three back to peg two. We move this peg to peg two, which is empty. After this m steps, the largest disk will remain unmoved at peg 1. We argue this by first moving the top 3 disks from peg one to peg three. To solve this puzzle with 4 disks, we can use or manipulate the result that we obtain from the previous puzzle with 3 disks.

Now, if there are 3 pegs and 3 + 1 = 4 disks, how many moves are needed? You need a total of 7 steps to complete the game. Let's look into a specific case where initially you have 3 pegs and 3 disks. Of course, a similar Tower of Hanoi with an additional disk will require more steps. Step 7: Move the smallest disk from peg one to peg three. Step 6: Move the middle-sized disk from peg two to peg three. Step 5: Move the smallest disk from peg two to the empty peg one. Step 4: Move the largest disk from peg one to the empty peg three. Step 3: Next, we take the smallest disk from peg three and put it on the top of the middle-sized disk at peg two.

Notice that we cannot place it at peg three since this violates the rule of the game, namely by placing a larger disk on a smaller one.

Step 2: Then, we take the middle-sized disk from peg one and move it to peg two. Step 1: We make our first move by taking the first disk (the smallest one) from peg one and move it to peg three. Suppose that we want to move the disks to the third peg. Now, we can move the 3 circular disks to either peg two or peg three, which are both empty. The first 3 denotes the 3 pegs and the second 3 means there are 3 circular disks. The game can be represented by the mathematical notation T(3,3).

Hanoi towers 4 pegs