A tromino or triomino is a

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

of order 3, that is, a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

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' ...

made of three equal-sized

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

s connected edge-to-edge.

Symmetry and enumeration

When

rotation 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

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s 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 In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

, 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

Geometrical dissection of an L-triomino (rep-4)

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-tile In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...

s. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- period ...

. 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 In geometry, a chair tiling (or L tiling) is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller copies and then re ...

. 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 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 Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...

, 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..

References

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 ...

{{Polyforms Polyforms

Symmetry and enumeration

Rep-tiling and Golomb's tromino theorem

See also

Previous and next orders

References

External links