Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some other
possibilities for yourself!

This problem is in two parts. The first part provides some building blocks which will help you to solve the final challenge. These can be attempted in any order. Of course, you are welcome to go straight to the Final Challenge without looking at the building blocks!

In this problem, you will be working on a famous mathematical puzzle called The Tower of Hanoi. There are three pegs, and on the first peg is a stack of discs of different sizes, arranged in order of descending size. The object of the game is to move all of the discs to another peg. However, only one disc can be moved at a time, and a disc cannot be placed on top of a smaller disc.

This interactivity shows the most efficient way of moving the discs from one end to the other:

Explain how you could work out the number of moves needed for the Tower of Hanoi puzzle with $n$ discs.

Extension:
There is a legend that a 64-disc version of the Tower of Hanoi is being played out in a temple, and when the final move is made, the world will come to an end. If one move is made each second, how long would it take to complete the game with 64 discs? Do we need to worry yet, if the first disc was moved at the very beginning of time?