A tromino is a $3 \times 1$ block:
What sized rectangles may be made using trominoes? You can print
off and cut out trominoes from this sheet
. Alternatively, if you
have a set of the game Jenga, then use the blocks as trominoes (but
check they are $3 \times 1$ first)
Can you cover $63$ squares of an $8\times 8$ chessboard using
trominoes? (Why can't you cover $64$?). If so, which square remains
uncovered, and are there other arrangements of the trominoes which
would lead to a different square being left uncovered?
Think of some other squares which cannot be covered with trominoes.
Can you "almost" cover them, leaving only one hole? When can an $n
\times n$ square be covered or "almost" covered?