# Tetromino
A tetromino is a polyomino of order 4, a connected plane figure formed by joining exactly four equal-sized squares edge to edge.[1](https://www.mathwords.com/t/tetromino.htm) Tetrominoes are best known as the seven falling pieces of the video game [Tetris](https://www.edgepedia.org/tetris), whose name combines the Greek word for four with tennis, Pajitnov's favorite sport.[2](https://www.mathsisfun.com/geometry/tetromino.html) • [3](https://time.com/2837390/tetris-at-30-pajitnov-interview/) In mathematics they are the fourth member of the polyomino family, after the monomino, domino, and tromino, and before the [pentomino](https://www.edgepedia.org/pentomino).[4](https://cimt.org.uk/resources/puzzles/pentoes/pentoint.htm)
| Key fact | Detail |
|---|---|
| Definition | Four unit squares joined along full edges; a polyomino of order 4[1](https://www.mathwords.com/t/tetromino.htm) |
| Count | 5 free tetrominoes, 7 one-sided tetrominoes, 19 fixed tetrominoes[5](https://mathworld.wolfram.com/Tetromino.html) |
| Free shapes | I, O, T, S, and L (the "math version", where mirror images count as one shape)[1](https://www.mathwords.com/t/tetromino.htm) |
| Tetris set | The 7 one-sided shapes I, O, T, J, L, Z, and S, since the game does not allow flipping pieces[2](https://www.mathsisfun.com/geometry/tetromino.html) |
| Naming origin | Solomon W. Golomb coined "polyomino" and its systematic names in a 1953 lecture to the Harvard Mathematical Club[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html) • [7](https://www.craftypuzzles.com/history/history-of-pentominoes.html) |
| Tetris connection | Alexey Pajitnov built Tetris in 1984 on an Electronika 60, adapting the pentomino puzzle game and reducing pieces to four squares[3](https://time.com/2837390/tetris-at-30-pajitnov-interview/) |
| T-tetromino tiling | An \( a \times b \) rectangle is tileable by the four T-tetromino variants if and only if \( 4 \mid a \) and \( 4 \mid b \)[8](https://www.math.ucla.edu/~pak/courses/285-Fall-2022/Lecture-notes.pdf) |
## The five shapes and their counts
Two tetrominoes that can be rotated or flipped to match each other count as the same free tetromino, giving exactly 5 distinct shapes: the I (straight), O (square), T, S (skew), and L shapes.[1](https://www.mathwords.com/t/tetromino.htm) If rotations are considered equivalent but reflections are not, the count is 7 one-sided tetrominoes, in which the mirror pairs J and L, and Z and S, are separated; this is the set used in Tetris, where pieces cannot be flipped over.[2](https://www.mathsisfun.com/geometry/tetromino.html) • [9](https://www.math.cmu.edu/~bkell/21110-2010s/polyominoes.html) If both rotations and reflections are treated as different, there are 19 fixed tetrominoes.[5](https://mathworld.wolfram.com/Tetromino.html) The same three-way counting scheme applies to polyominoes generally.[9](https://www.math.cmu.edu/~bkell/21110-2010s/polyominoes.html)
The full set can be constructed by starting with a single square and adding squares one at a time, each new square sharing a full edge with the growing shape.[1](https://www.mathwords.com/t/tetromino.htm) Unlike the counts for small orders, there is no general formula for the number of distinct polyominoes with \( n \) squares; the only method is explicit construction from the polyominoes with \( n - 1 \) squares.[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html)
## Origin of the term
The American mathematician [Solomon W. Golomb](https://www.edgepedia.org/solomon-w-golomb) (1932–2016), a longtime [University of Southern California](https://www.edgepedia.org/university-of-southern-california) faculty member, introduced the polyomino names in a 1953 lecture to the Harvard Mathematical Club, and his work was published in the *American Mathematical Monthly* the following year.[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html) • [10](https://hub.jhu.edu/2016/05/04/solomon-golomb-obit/) "Polyomino" is a back-formation from "domino", with the prefix indicating the number of squares; Golomb's systematic names include tromino, tetromino, and pentomino.[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html) • [7](https://www.craftypuzzles.com/history/history-of-pentominoes.html) [Martin Gardner](https://www.edgepedia.org/martin-gardner)'s "Mathematical Games" column in *Scientific American* in May 1957 carried the shapes to a wide audience, and Golomb's 1965 book *Polyominoes*, whose first edition was bundled with a set of the twelve pentomino pieces, became a classic of recreational mathematics.[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html) • [11](https://puzzlebyrinth.com/en/articles/pentomino-1965) Golomb received the National Medal of Science in 2013.[10](https://hub.jhu.edu/2016/05/04/solomon-golomb-obit/)
## Tetris
Tetris grew out of Golomb's shapes. [Alexey Pajitnov](https://www.edgepedia.org/alexey-pajitnov), working in the Soviet Union, knew pentominoes as a favorite puzzle game and translated the idea onto an Electronika 60 computer, using symbols on screen because of the machine's limitations. He reduced the five-square pentomino pieces to four-square tetrominoes, named the game after the Greek word for four, and settled on the rule that a completed row disappears: "drop the pieces down, and when a row was filled, remove it because there was no reason to keep it on the screen".[3](https://time.com/2837390/tetris-at-30-pajitnov-interview/) The game uses the 7 one-sided tetrominoes, since mirror-image pieces are distinct when flipping is not allowed.[2](https://www.mathsisfun.com/geometry/tetromino.html) Golomb's polyominoes are described as the inspiration for Tetris in accounts of his career.[10](https://hub.jhu.edu/2016/05/04/solomon-golomb-obit/)
## Tiling mathematics
Tetrominoes support a body of enumerative and computational work. A classical question asks which rectangles a given tetromino can tile. For the T-tetromino, an \( a \times b \) rectangle can be tiled by the four T-tetromino variants if and only if both \( a \) and \( b \) are divisible by 4.[8](https://www.math.ucla.edu/~pak/courses/285-Fall-2022/Lecture-notes.pdf) Coloring arguments, in which a board's squares are colored and the counts of colors covered by each tile are compared, can prove that certain tilings are impossible; the same technique shows, for example, that a chessboard with two white corner squares removed cannot be covered by dominoes.[6](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html)
The counts grow quickly. A tetromino tiling of a 10 × 10 board has 7,213,560,548,906,621 distinct solutions, and a 12 × 12 board has 13,664,822,582,333,502,156,627,512 solutions; storing the 10 × 10 solution set naively would require about 100 petabytes, while a zero-suppressed binary decision diagram represents it with 16,476,396 nodes, about 300 megabytes.[12](https://ntt-review.jp/archive/ntttechnical.php?contents=ntr201711fa3.html) Deciding whether a given polyomino can tile a finite region is NP-complete for the L-tromino, whereas tiling with the square (2 × 2) tetromino is solvable in polynomial time, and the general polyomino tiling problem is NP-complete, meaning no polynomial-time algorithm is known, although a proposed tiling can be checked quickly.[13](https://andrewwinslow.com/papers/polyprobs-dlt18.pdf) • [14](https://www.sciencedirect.com/science/article/abs/pii/S187775032500211X) • [15](https://arxiv.org/abs/math/0003039)
## References
1. [Tetromino — Definition, Formula & Examples, Mathwords](https://www.mathwords.com/t/tetromino.htm)
2. [The Set of Tetrominoes, Math is Fun](https://www.mathsisfun.com/geometry/tetromino.html)
3. [Tetris at 30: An Interview with the Historic Puzzle Game's Creator, TIME](https://time.com/2837390/tetris-at-30-pajitnov-interview/)
4. [Pentominoes — Introduction, CIMT](https://cimt.org.uk/resources/puzzles/pentoes/pentoint.htm)
5. [Tetromino, Wolfram MathWorld](https://mathworld.wolfram.com/Tetromino.html)
6. [Polyominoes, ETH Library](https://library.ethz.ch/en/collections-and-archives/platforms/virtual-exhibitions/Its-all-math-and-games/polyominoes.html)
7. [History of Pentominoes and Polyomino Puzzles, Crafty Puzzles](https://www.craftypuzzles.com/history/history-of-pentominoes.html)
8. [Tiling by T-tetrominoes, lecture notes, UCLA](https://www.math.ucla.edu/~pak/courses/285-Fall-2022/Lecture-notes.pdf)
9. [21-110: Polyominoes, Carnegie Mellon University](https://www.math.cmu.edu/~bkell/21110-2010s/polyominoes.html)
10. [Solomon Golomb, pioneering scholar of mathematics and engineering, dies at 83, Johns Hopkins Hub](https://hub.jhu.edu/2016/05/04/solomon-golomb-obit/)
11. [Pentomino (1965) — How Solomon Golomb's Five-Square Shape Became Tetris, Puzzlebyrinth](https://puzzlebyrinth.com/en/articles/pentomino-1965)
12. [Efficient Algorithm for Enumerating All Solutions to an Exact Cover Problem, NTT Technical Review](https://ntt-review.jp/archive/ntttechnical.php?contents=ntr201711fa3.html)
13. [Visual elements on polyomino tiling problems, Andrew Winslow (DLT 2018)](https://andrewwinslow.com/papers/polyprobs-dlt18.pdf)
14. [A generalized colouring method for a parallelizable integer linear programming approach to polyomino tiling, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S187775032500211X)
15. [Hard Tiling Problems with Simple Tiles](https://arxiv.org/abs/math/0003039)
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*Topic: Encyclopedia › Sports, games, and recreation › Board, card, and puzzle games › Puzzles › Physical, logic, and word puzzles › Tiling puzzles and polyominoes*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 21, 2026 · Edited: Sep 30, 2026; Oct 21, 2026 · Last review: Oct 21, 2026*
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