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 popu ...

of size 3, that is, a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

in the

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

made of three equal-sized

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s connected edge-to-edge.

Symmetry and enumeration

When

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

s and

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, mirror-like reflection of waves from a surface *** Mirror image, a reflection in a mirror or in water ** Diffuse r ...

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 (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

, 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-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 Tessellation, 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 set of prototiles, aperiodic if copie ...

. 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 resc ...

. 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 domin ...

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. He most notably inven ...

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 mathematical proof, 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), \dots all hold. This is done by first proving a ...

, 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 Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

Tromino Puzzle
at cut-the-knot

at

Amherst College Amherst College ( ) is a Private college, private Liberal arts colleges in the United States, liberal arts college in Amherst, Massachusetts, United States. Founded in 1821 as an attempt to relocate Williams College by its then-president Zepha ...

{{Polyforms Polyforms

Symmetry and enumeration

Rep-tiling and Golomb's tromino theorem

See also

Previous and next orders

References

External links