tromino

When rotations and reflections are not considered to be distinct shapes, there are only two different free trominoes: "I" and "L" (the "L" shape is also called "V").

Since both free trominoes have reflection symmetry, they are also the only two one-sided trominoes (trominoes with reflections considered distinct). When rotations are also considered distinct, there are six fixed trominoes: two I and four L shapes. They can be obtained by rotating the above forms by 90°, 180° and 270°.{{mathworld|title=Triomino|id=Triomino}}{{cite journal |last=Redelmeier |first=D. Hugh |year=1981 |title=Counting polyominoes: yet another attack |journal=Discrete Mathematics |volume=36 |issue=2 |pages=191–203 |doi=10.1016/0012-365X(81)90237-5|doi-access=free }}

File:Geometrical dissection of an L-triomino (rep-4).gif

Both types of tromino can be dissected into n² smaller trominos of the same type, for any integer n > 1. That is, they are rep-tiles.{{citation

| last = Nițică | first = Viorel

| contribution = Rep-tiles revisited

| location = Providence, RI

| mr = 2027179

| pages = 205–217

| publisher = American Mathematical Society

| title = MASS selecta

| year = 2003}}. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.{{cite journal

| last = Robinson | first = E. Arthur Jr.

| doi = 10.1016/S0019-3577(00)87911-2

| issue = 4

| journal = Indagationes Mathematicae

| mr = 1820555

| pages = 581–599

| title = On the table and the chair

| volume = 10

| year = 1999| doi-access = free

}}.

Motivated by the mutilated chessboard problem, Solomon W. Golomb used this tiling as the basis for what has become known as Golomb's tromino theorem: if any square is removed from a 2ⁿ × 2ⁿ chessboard, the remaining board can be completely covered with L-trominoes. To prove this by mathematical induction, partition the board into a quarter-board of size 2ⁿ⁻¹ × 2ⁿ⁻¹ that contains the removed square, and a large tromino formed by the other three quarter-boards. The tromino can be recursively dissected into unit trominoes, and a dissection of the quarter-board with one square removed follows by the induction hypothesis.

In contrast, when a chessboard of this size has one square removed, it is not always possible to cover the remaining squares by I-trominoes.{{cite journal

| last = Golomb | first = S. W. | author-link = Solomon W. Golomb

| doi = 10.2307/2307321

| journal = American Mathematical Monthly

| mr = 0067055

| pages = 675–682

| title = Checker boards and polyominoes

| volume = 61

| year = 1954| issue = 10 | jstor = 2307321 }}.

• Domino
• Tetromino

{{reflist}}

• [http://www.cut-the-knot.org/Curriculum/Geometry/Tromino.shtml Golomb's inductive proof of a tromino theorem] at cut-the-knot
• [http://www.cut-the-knot.org/Curriculum/Games/TrominoPuzzle.shtml Tromino Puzzle] at cut-the-knot
• [http://www.amherst.edu/~nstarr/puzzle.html Interactive Tromino Puzzle] at Amherst College

{{Polyforms}}

Category:Polyforms