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A tromino or triomino is a polyomino of order 3, that is, a polygon in the
Both types of tromino can be dissected into ''n''2 smaller trominos of the same type, for any integer ''n'' > 1. That is, they are rep-tiles. 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
Golomb's inductive proof of a tromino theorem
at
Tromino Puzzle
at cut-the-knot
at
A tromino or triomino is a polyomino of order 3, that is, a polygon in the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
made of three equal-sized squares connected edge-to-edge.
Symmetry and enumeration
Whenrotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s 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°.
Rep-tiling and Golomb's tromino theorem
Both types of tromino can be dissected into ''n''2 smaller trominos of the same type, for any integer ''n'' > 1. That is, they are rep-tiles. 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
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks:
Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoe ...
, Solomon W. Golomb
Solomon Wolf Golomb (; May 30, 1932 – May 1, 2016) was an American mathematician, engineer, and professor of electrical engineering at the University of Southern California, best known for his works on mathematical games. Most notably, he inve ...
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..
See also
Previous and next orders
* Domino * TetrominoReferences
External links
Golomb's inductive proof of a tromino theorem
at
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Tromino Puzzle
at cut-the-knot
at
Amherst College
Amherst College ( ) is a private liberal arts college in Amherst, Massachusetts. Founded in 1821 as an attempt to relocate Williams College by its then-president Zephaniah Swift Moore, Amherst is the third oldest institution of higher educatio ...
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