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.
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.
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan.
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
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.
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. Some tiling puzzles ask players to dissect a given shape first and then rearrange the pieces into another shape. Other tiling puzzles ask players to dissect a given shape while fulfilling certain conditions. The two latter types of tiling puzzles are also called dissection puzzles.
The 3-partition problem is a strongly NP-complete problem in computer science. The problem is to decide whether a given multiset of integers can be partitioned into triplets that all have the same sum. More precisely:
In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
The Eternity II puzzle is an edge-matching puzzle launched on 28 July 2007. It was developed by Christopher Monckton and marketed and copyrighted by TOMY UK Ltd as a successor to the original Eternity puzzle. The puzzle was part of a competition in which a $2 million prize was offered for the first complete solution. The competition ended at noon on 31 December 2010, with no solution being found.
A three-dimensional edge-matching puzzle is a type of edge-matching puzzle or tiling puzzle involving tiling a three-dimensional area with polygonal pieces whose edges are distinguished with colors or patterns, in such a way that the edges of adjacent pieces match. Edge-matching puzzles are known to be NP-complete, and capable of conversion to and from equivalent jigsaw puzzles and polyomino packing puzzle.
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.
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.
Serpentiles is the name coined by Kurt N. Van Ness for the hexagonal tiles used in various edge-matching puzzle connection abstract strategy games, such as Psyche-Paths, Kaliko, and Tantrix. For each tile, one to three colors are used to draw paths linking the six sides together in various configurations. Each side is connected to another side by a specific path route and color. Gameplay generally proceeds so that players take turns laying down tiles. During each turn, a tile is laid adjacent to existing tiles so that colored paths are contiguous across tile edges.
In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other words, it asks whether the variables of a given Boolean formula—whose incidence graph consisting of variables and clauses can be embedded on a plane—can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
In discrete mathematics and theoretical computer science, reconfiguration problems are computational problems involving reachability or connectivity of state spaces.
Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap. Several variants of this problem have been studied.
MacMahon Squares are an edge-matching puzzle first published by Percy MacMahon in 1921, using 24 unique squares with 3-color patterns; each of the four edges is assigned a single color. The complete set of 24 squares are organized next to each other by matching edge colors to create a 4 by 6 grid. Such tessellation puzzles have multiple variants, which are determined by restrictions on how to arrange the 24 squares. This game has also been commercialized in numerous physical forms, by various companies.
