A sheet of paper is torn into 3 pieces. Then, one of these pieces is torn again into 3 pieces, and so on. Can we obtain, at some point, a total of 100 pieces? 2017 pieces?
We start with an odd number (1) of parts. In each operation, we divide one of the parts into 3, which adds two new parts, keeping the number of parts odd (our invariant). It is therefore impossible to obtain a total of 100 parts. However, by repeating the operation 1008 times, we do end up with 2017 parts.