Katamino is the puzzle of placing twelve polygonal pieces so as to form a $5\times 12$ rectangular array. The pieces consist of all possible arrangements of five connected $1\times 1$ squares. For example, there is the $5\times 1$ piece, the T piece, and so forth.
Here is one possible solution of the Katamino puzzle:
Question. How many Katamino solutions are there on a $5 \times 12$ board? Answers are also welcome for other sized boards (but specify the piece set).
I know that there must be at least four ways to arrange the pieces because I have successfully arranged them in one way, and this arrangement can be flipped horizontally, vertically, or in both directions to create new arrangements. For the same reason, because of this symmetry, the total number of arrangements must be a multiple of four. But how many solutions are there?
There are 1010 such tilings. The key term here is polyominoes; for the case of five squares glued along their edges, these are called pentominoes. Googling will yield lots more; see also the classic book of SW Golomb.
For this particular question, see e.g. here for an enumeration of all of them.