59th Ukrainian National Mathematical Olympiad

We will cover the $1\times 1$ cells two ways as shown in Fig. 13. Since any $3\times 1$ rectangle covers exactly one cell of each color, only white cell can be left uncovered (since there are 6 white and 5 of grey and black cells). Same positions of white cells are only on the edges of $4\times 4$ square. Thus, there are 4 kinds of uncovered cell. Clearly, the number of covers for each of these types is the same. It is easy to find that there are four covers (puc. 14). There are three ways with one horizontal $3\times 1$ rectangle on the bottom, and one way without it. Thus, there are 16 ways in total.