22. (a) Color the first, third, and fifth row red, and the remaining squares white. There in total $n$ pieces and $3 n$ red squares. Since each piece can cover at most three red squares, it follows that each piece colors exactly three red squares. Then it follows that the two white squares it covers must be on the same row; otherwise, the piece has to cover at least three. Hence, each white row can be partitioned into pairs of squares belonging to the same piece. Thus it follows that the number of white squares in a row, which is $n$, must be even. (b) Let $a_{k}$ denote the number of different tilings of a $5 \times 2 k$ rectangle. Let $b_{k}$ be the number of tilings that cannot be partitioned into two smaller tilings along a vertical line (without cutting any pieces). It is easy to see that $a_{1}=b_{1}=2, b_{2}=2, a_{2}=6=2 \cdot 3, b_{3}=4$, and subsequently, by induction, $b_{3 k} \geq 4, b_{3 k+1} \geq 2$, and $b_{3 k+2} \geq 2$. We also have $a_{k}=b_{k}+\sum_{i=1}^{k-1} b_{i} a_{k-i}$. For $k \geq 3$ we now have inductively $$a_{k}>2+\sum_{i=1}^{k-1} 2 a_{k-i} \geq 2 \cdot 3^{k-1}+2 a_{k-1} \geq 2 \cdot 3^{k}$$