A **tiling with rectangles** is a tiling which uses rectangles as its parts. The domino tilings are tilings with rectangles of 1 × 2 side ratio. The tilings with straight polyominoes of shapes such as 1 × 3, 1 × 4 and tilings with polyominoes of shapes such as 2 × 3 fall also into this category.

Some tiling of rectangles include:

Stacked bond | Running bond | Basket weave | Basket weave | Herringbone pattern |

The smallest square that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 11 × 11 square, and the tiling uses five rectangles.^{ [1] }

The smallest rectangle that can be cut into (m × n) rectangles, such that all m and n are different integers, is the 9 × 13 rectangle, and the tiling uses five rectangles.^{ [1] }^{ [2] }

Derived from the Greek word for '5', and "domino", a **pentomino** is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different *free* pentominoes. When reflections are considered distinct, there are 18 *one-sided* pentominoes. When rotations are also considered distinct, there are 63 *fixed* pentominoes.

**Squaring the square** is the problem of tiling an integral square using only other integral squares. The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be **perfect**, meaning the sizes of the smaller squares are all different. A related problem is **squaring the plane**, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

In Euclidean plane geometry, a **rectangle** is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a *square*. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices *ABCD* would be denoted as *ABCD*.

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.

**Packing problems** are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.

A **tessellation** or **tiling** is the covering of a surface, often a plane, using one or more geometric shapes, called *tiles*, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

A **hexomino** is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different *free* hexominoes. When reflections are considered distinct, there are 60 *one-sided* hexominoes. When rotations are also considered distinct, there are 216 *fixed* hexominoes.

**1000** or **one thousand** is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: **1,000**.

In recreational mathematics, a **polyabolo** is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gardner in his June 1967 "Mathematical Games column" in *Scientific American*.

A **polycube** is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.

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.

A **heptomino** is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hept(a)-. When rotations and reflections are not considered to be distinct shapes, there are 108 different *free* heptominoes. When reflections are considered distinct, there are 196 *one-sided* heptominoes. When rotations are also considered distinct, there are 760 *fixed* heptominoes.

A **nonomino** is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different *free* nonominoes. When reflections are considered distinct, there are 2,500 *one-sided* nonominoes. When rotations are also considered distinct, there are 9,910 *fixed* nonominoes.

**216** is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato.

In mathematics, the **plastic number***ρ* is a mathematical constant which is the unique real solution of the cubic equation

**277** is the natural number following 276 and preceding 278.

In geometry, a **domino tiling** of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.

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 Martin Gardner in his "Mathematical Games" column in the May 1963 issue of *Scientific American*. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in *Mathematics Magazine*.

In geometry, a **partition** of a polygon is a set of primitive units, which do not overlap and whose union equals the polygon. A **polygon partition problem** is a problem of finding a partition which is minimal in some sense, for example a partition with a smallest number of units or with units of smallest total side-length.

* Polyominoes: Puzzles, Patterns, Problems, and Packings* is a mathematics book on polyominoes, the shapes formed by connecting some number of unit squares edge-to-edge. It was written by Solomon Golomb, and is "universally regarded as a classic in recreational mathematics". The Basic Library List Committee of the Mathematical Association of America has strongly recommended its inclusion in undergraduate mathematics libraries.

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