Tiling, area, and square numbers

Cover the board with given tile shapes and relate the counts to area. A tile of $s$ squares can cover the board only if the total, $64$, is a multiple of $s$: a $1 \times 2$ domino ($s = 2$) can tile it, a straight tromino ($s = 3$) cannot, since $3$ does not divide $64$. Area gives a fast impossibility test.

Square numbers appear when the tiles are squares: covering an $n \times n$ area with unit tiles needs $n^2$ of them, and mixing square tiles of different sizes writes $64$ as a sum of squares. Divisibility rules out many tile sets before you place a single piece.