Chapter 5.3, Problem 23E

The Towers of Hanoi is a popular puzzle. It consists of three pegs and a number of discs of differing diameters, each with a hole in the center. The discs initially sit on one of the pegs in order of decreasing diameter (smallest at top, largest at bottom, as in Fig. 5.5), thus forming a triangular tower. The object is to move the tower to one at a time in such a way that no disc is ever placed upon a smaller one.

1. Solve the puzzle when there are $n=2$ discs and show your moves by completing a little table like that below. [The pegs are labeled A, B, C, and we have used an asterisk (*) to denote an empty peg. The disks are numbers in order of increasing size, thus disk 1 is the smallest.]

   $\begin{array}{cccc}& A& B& C\\ \text{Intial position}& 1,2& *& *\\ \text{Move 1}& ??& ??& ??\\ \text{Move 2}& ??& ??& ??\\ \text{etc}.& & & \end{array}$

   Also solve the puzzle, with a similar table, when $n=3$. How many moves are required in each case?

2. Give a recurrence relation for $a_n$, the number of moves required to transfer $n$ discs from one peg to another.

3. Find an explicit formula for $a_n$.

4. Suppose we can move a disc a second. Estimate the time required to transfer the discs if $n=8,n=16,n=32,$ and $n=64$.