Make a table of the 2n elements of the Gray code for some n, by writing them in order starting from 0n, one underneath the other.

For example, when n = 3

000

001

011

010

110

111

101

100

a. Prove by induction that each successive Gray code changes the last (rightmost) bit, or else the bit to the left of the rightmost 1.

b. Look at the columns of your Gray codes and conjecture a theorem regarding the longest consecutive sequence of 1's in the ith column from the right.

c. Prove your theorem by induction.