2. (20 points) A unit segment is divided into 3 equal parts, and the middle part is discarded. Each of the remaining two segments is in turn divided into 3 equal parts, and its middle part is also discarded, after which the same operation is performed on each of the remaining segments, and so on. Suppose the operation is repeated 16 times. How many segments of length $1 / 3^{16}$ remain?
Answer: $2^{16}$.

Solution: Notice that after each operation, instead of one original segment, there remain two segments, each of which is one-third the length of the original. Thus, after each operation, the number of remaining segments doubles. Therefore, after sixteen operations, there will be $2^{16}$ segments remaining, and the length of each will be $1 / 3^{16}$.