Fill in any one square, then cover the rest of the board with L-shaped pieces.
Or click any square on the board to move the filled-in square there. You can move it until you place your first piece.
Can it always be done, no matter which square is filled in, and on boards of every size? Try a few boards before you look.
It doesn't matter which square you fill in. Every board with 2n squares on a side can be finished, and here is the argument.
Solomon Golomb proved this in 1954, in the same paper where he introduced the word “polyomino” for shapes made of squares joined edge to edge. His proof is an early classic of proof by induction: the 8 × 8 case rests on the 4 × 4 case, which rests on the 2 × 2 case.
In memory of Norton Starr, mathematics professor at Amherst College, who commissioned Kadon Enterprises to make VEE-21, a set of 21 of these pieces, to demonstrate a theorem. Twenty-one is exactly enough to cover an 8 × 8 board with one square filled in.