22. C2 (ARM) For $n$ an odd positive integer, the unit squares of an $n \times n$ chessboard are colored alternately black and white, with the four corners colored black. A tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with nonoverlapping trominos? When it is possible, what is the minimum number of trominos needed?
22. Write $n=2 k+1$. Consider the black squares at an odd height: there are $(k+1)^{2}$ of them in total and no two can be covered by one trimino. Thus, we always need at least $(k+1)^{2}$ triminoes, which cover $3(k+1)^{2}$ squares in total. However, $3(k+1)^{2}$ is greater than $n^{2}$ for $n=1,3,5$, so we must have $n \geq 7$.
The case $n=7$ admits such a covering, as shown in Figure 1. For $n>7$ this is possible as well: it follows by induction from Figure 2 .