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**Tiling puzzles** are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask you to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles.

Tiling puzzles may be made from wood, metal, cardboard, plastic or any other sheet-material. Many tiling puzzles are now available as computer games.

Tiling puzzles have a long history. Some of the oldest and most famous are jigsaw puzzles and the tangram puzzle.

Other examples of tiling puzzles include:

- Conway puzzle
- Domino tiling, of which the mutilated chessboard problem is one example
- Eternity puzzle
- Geometric magic square
- Puzz-3D
- Squaring the square
- Tantrix
- T puzzle

Many three-dimensional mechanical puzzles can be regarded as three-dimensional tiling puzzles.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

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.

A **jigsaw puzzle** is a tiling puzzle that requires the assembly of often oddly shaped interlocking and mosaiced pieces, each of which typically has a portion of a picture; when assembled, they produce a complete picture.

**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.

In geometry, the **Dehn invariant** of a polyhedron is a value used to determine whether polyhedra can be cut into pieces and reassembled into each other or whether they can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other.

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** is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.

A **dissection puzzle**, also called a **transformation puzzle** or *Richter Puzzle*, is a tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes. The creation of new dissection puzzles is also considered to be a type of dissection puzzle. Puzzles may include various restraints, such as hinged pieces, pieces that can fold, or pieces that can twist. Creators of new dissection puzzles emphasize using a minimum number of pieces, or creating novel situations, such as ensuring that every piece connects to another with a hinge.

A **sliding puzzle**, **sliding block puzzle**, or **sliding tile puzzle** is a combination puzzle that challenges a player to slide pieces along certain routes to establish a certain end-configuration. The pieces to be moved may consist of simple shapes, or they may be imprinted with colours, patterns, sections of a larger picture, numbers, or letters.

In geometry, a **dissection problem ** is the problem of partitioning a geometric figure into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a **dissection**. It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox and Tarski's circle-squaring problem, the pieces are typically required to be well-behaved. For instance, they may be restricted to being the closures of disjoint open sets.

In geometry, a **tile substitution** is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.

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, an **equidissection** is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triangles. In fact, most polygons cannot be equidissected at all.

A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The *aperiodicity* referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.

A **self-tiling tile set**, or *setiset*, of order *n* is a set of *n* shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of *n* shapes. That is, the *n* shapes can be assembled in *n* different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for *n* = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012, but the problem of finding such sets for *n* = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes and polyominoes were previously published by Gardner.

In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every convex polytope could be dissected into orthoschemes.

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