
The game called "New Tetris", for Nintendo 64 lets a player drop Tetris pieces, which are 4celled polyominoes, or tetrominoes, into a rectangular playing field. The objective is to complete "rows" by densely packing the pieces so that all 10 cells in a given row are covered by a tetromino. Higher scores can be achieved by first packing the tetrominoes into perfect 4by4 squares, which glow gold (when packed with four identical tetrominoes) or silver (otherwise), and then afterwards completing the rows that contain the gold or silver 4by4 squares. In playing the game, I was struck by the wide variety of combinations of Tetris pieces that can be used to form gold and silver squares. By hand, I drew diagrams of 35 such tilings, and then I designed an Excel spreadsheet that found two others, for a total of 37 — 8 golden tilings, and 29 silver ones.
This page describes the ways to tile an nbyn square with onesided polyominoes with n cells (npolyominoes). Two tilings are considered distinct if neither one is a rotation of the other. OEIS doesn't seem to contain the sequence of the numbers of such ntilings for n=1, 2, 3, 4, ... This sequence is 1, 1, 3, 37, ...
To be sure that 37 is the correct number of distinct tilings of a 4by4 square by Tetris pieces, I designed an Excel spreadsheet that enumerated all 113 ways to position one of the 29 fixed tetrominoes within a 4by4 square. I call each of these 113 ways a "positioning". Then I superimposed all 113*112/2 distinct pairs of positionings to find those without any overlapping. Finally, I matched up those pairs of positionings with other pairs that completely filled the square. I then eliminated all but one of each tiling that can be obtained by rotating other tilings to arrive at 37 tilings.
Section 1: Enumeration of onesided npolyominoes, and diagrams of all distinct tilings of nbyn squares by these pieces, for n=0,1,2,3,4, and maybe someday, 5.
Section 2: Detailed description of the method of finding distinct tilings of an nbyn square by npolyominoes, for n=0,1,2,3,4, and 5.
Section 3: Odds and Ends: Interesting factoids that arose in connection with this topic.
This section enumerates onesided npolyominoes, and diagrams of all distinct tilings of nbyn squares by these pieces, for n=0,1,2,3,4, and maybe someday, 5.
What? You didn't see it? It's there, trust me, just above this paragraph, and below the title. It's just too small to see. Just as there is exactly one null set, there is exactly one 0celled onesided polyomino.
But the number of such polyominoes required to tile a 0by0 square is arbitrary. For example, you can easily fit five 0celled polyominoes inside a 0by0 square without overlapping them (except at their borders).
So the n=0 case is undefined. I'm really sorry to start Section 1 with such a deep exercise in abstract thinking. Don't worry, it gets much better.
There is just one 1celled polyomino, and one way to use it to tile a 1by1 square:
1celled polyomino  tilings of the 1by1 square 
There is just one 2celled polyomino, and one way to use it to tile a 2by2 square:
2celled polyomino  tilings of the 2by2 square 
Remember, tilings are considered distinct only if none can be obtained by rotating another.
There are two 3celled polyominoes, and three ways to use them to tile a 3by3 square:
3celled polyominoes  tilings of the 3by3 square  
The distinction between onesided and free polyominoes isn't important for 1, 2, and 3celled polyominoes because all of these small pieces have bilateral symmetry. This distinction becomes important only for 4celled and larger polyominoes. There are five 4celled free polyominoes. Two of them lack bilateral symmetry, so their mirror images form two of the seven 4celled onesided polyominoes.
There are seven 4celled onesided polyominoes:
4celled onesided polyominoes  

tilings of the 4by4 square  

The sequence of the number of ways to tile an nbyn square with onesided polyominoes with n cells (npolyominoes) for n=1,2,3,4,... is 1,1,3,37,... Note: two tilings are considered distinct if neither one is a rotation of the other.
Section 2: Detailed description of the method of finding distinct tilings of an nbyn square by npolyominoes, for n=0,1,2,3,4, and maybe someday, 5.
This section is currently under construction . . . . . .
(an overview will be created later.)
Odds and ends are interesting factoids that arose in connection with this topic.
MerriamWebster defines chiral as: of, relating to, or being a molecule that is nonsuperimposable on its mirror image. Of polyominoes, the word is used to describe ones that are distinct — or considered distinct — from their mirror images. OEIS applies the word to the sequence of numbers of onesided polyominoes with n cells to mean that some of the polyominoes for each n are chiral in that they are considered distinct from their mirror images. The description of A000988 is: Number of onesided (or chiral) polyominoes with n cells. When n is 4, for example, there are two pairs of chiral tetrominoes: L,J and Z,S. Together with the I, O, and T, there are 7 onesided tetrominoes. Compare this to A000105, the Number of polyominoes with n cells, which considers L and J to be the same, and Z and S to be the same. Viewed this way, there are five tetrominoes: L, Z, I, O, and T. None of them are chiral, because none of them is considered distinct from its mirror image.
I think the polyomino entries in OEIS should be changed as follows.
A000988, the number of ncelled onesided polyominoes, uses the word "chiral" in a misleading way. Chiral means distinct from its mirror image. True, it is reasonable to characterize free polyominoes (A000105) as nonchiral in that no free polyomino is considered distinct from its mirror image. However not all onesided polyominoes (A000988) are chiral. In fact, a better sequence to describe chiral polyominoes is A030228, which are the polyominoes lacking bilateral symmetry.
I suggest removing the word "chiral" from the description of A000988, and adding the following comment to all three sequences, along with appropriate crossreferences (I checked later, and found these suggestions were adopted):
A000105(n) + A030228(n) = A000988(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of onesided polyominoes.
Tetriswiki: Square_Platforming
OEIS: A000105 — number of polyominoes with n cells  1, 1, 1, 2, 5, 12, ...
OEIS: A000988 — number of onesided polyominoes with n cells  1, 1, 1, 2, 7, 18, ...
Wikipedia: Polyomino is a clear description of polyominoes — free, onesided, and fixed.
Wikipedia: Tetromino describes 4celled polyominoes — the onesided ones are the seven Tetris pieces.
Polyominoes enumerates and classifies polyominoes by their bilateral and rotational symmetries, and the size of the smallest rectangle that encloses them.
Tetromino Soup is a puzzle asking for the probability of forming each of the five free tetrominoes by a random walk
The webmaster and author of this Math Help site is Graeme McRae.