The Towers of Hanoi puzzle (Figure 25.10) is to move the stack of disks on peg 1 to peg 2, while…

The Towers of Hanoi puzzle (Figure 25.10) is to move the stack of disks on peg 1 to peg 2, while obeying the following constraints: you can move only one disk at a time from peg to peg; and you can never put a bigger disk on top of a smaller disk.

(a) Describe a recursive algorithm for solving the puzzle. How many steps does it take, if there are n disks?

(b) Show that an n-bit Gray code also solves the problem. Simply run through the codes in order, at each step moving the disk corresponding to the bit position that changes (where disk 0 is the smallest disk, and bit position 0 is the rightmost.) If the disk to be moved is any but the smallest, there is only one peg to which it can legally be moved. The smallest disk moves cyclically, either 2→3→1→2→… if n is odd, or 3→2→1→3→… if n is even.

The Towers of Hanoi puzzle (Figure 25.10) is to move the stack of disks on peg 1 to peg 2, while...

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