DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

a way that each n-string differs from the previous one in exactly
one place. Describe how to get your Gray Code for 3-strings from
some Gray code for 2-strings.
(b) Can you describe how to get some Gray Code for 4-strings from
the one you found for 3-strings?
(c) Give a written description of the inductive step of an inductive
proof that of the existence of Gray codes for n-strings for all n ≥ 1.
One of the original uses of Gray codes was to reduce coding errors in a
pulse communication system. Frank Gray of Bell Labs received a patent for
Gray codes2 in 1953, but this type of sequence is now known to have been
used even earlier (in the 1870s) in a telegraph device by Émile Baudot, a
French engineer.
→64. Use the definition of a Gray Code to find a bijection between the evensized
subsets of [n] and the odd-sized subsets of [n].
Hint. Consider the characteristic functions of subsets.
Figure 2.1: The Towers of Hanoi Puzzle
•65. The Towers of Hanoi Puzzle (refer to Figure 2.1) has three rods rising
from a rectangular base and n ≥ 1 rings of different sizes. At the
beginning of the puzzle all rings are stacked on one rod in order of
decreasing size. A legal move consists of moving a ring from one rod
to another in such a way that the ring does not land on a smaller one.
Let mn be the (minimal) number of moves required to move all the
rings from the initial rod to any other rod. Find mn for all n ≤ 3 and
use those solutions to develop a strategy for obtaining the number of
moves for N + 1 rings from the number of moves for N rings. Describe
this as an inductive process. Guess a formula for mn and use induction
to show your guess is correct.
2 Pulse Code Communication, United States Patent Number 2632058 (March 17, 1953)
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