Tromino

 ( Polyforms ) 

A tromino or triomino is a polyomino of size 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.

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Symmetry and enumeration When and reflections are not considered to be distinct shapes, there are only two different free polyomino 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°.

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Rep-tiling and Golomb's tromino theorem Both types of tromino can be dissected into n² smaller trominos of the same type, for any integer n > 1. That is, they are .. 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..

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 2 ⁿ × 2 ⁿ 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 2 ⁿ⁻¹ × 2 ⁿ⁻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..

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See also

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Previous and next orders

• Domino
• Tetromino

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External links

• Golomb's inductive proof of a tromino theorem at cut-the-knot
• Tromino Puzzle at cut-the-knot
• Interactive Tromino Puzzle at Amherst College

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