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Polyomino

A polyomino is a plane geometric figure formed by joining a finite number of equal-sized squares edge to edge, resulting in a connected shape.^[1] These shapes, known as n-ominoes when composed of n squares, generalize the domino (a 2-omino) and are fundamental objects in recreational mathematics and combinatorics.^[2] The term "polyomino" was coined by mathematician Solomon W. Golomb in 1953, building on earlier puzzles dating back to the early 20th century, and was popularized through Martin Gardner's "Mathematical Games" columns in Scientific American starting in 1957.^[3]^[1] Polyominoes are classified into three main types based on symmetries: free polyominoes, which consider rotations and reflections as identical; one-sided polyominoes, which treat reflections as distinct but allow rotations; and fixed polyominoes, which distinguish all orientations.^[1] For small n, the numbers of free polyominoes are well-known: 1 monomino, 1 domino, 2 trominoes, 5 tetrominoes, and 12 pentominoes.^[1] Enumeration of polyominoes remains an active area of research, with exact counts computed up to n=70 for fixed polyominoes (as of 2024)^[4] and asymptotic growth rates estimated around a constant of approximately 4.06.^[2] Beyond counting, polyominoes feature prominently in tiling problems, where sets like the 12 pentominoes are used to cover rectangles or other regions without gaps or overlaps, and in puzzles that explore their packing and dissection properties.^[3] They also appear in popular culture, notably as the falling "tetriminoes" in the video game Tetris, which employs the seven one-sided tetrominoes.

Fundamentals

Definition and Nomenclature

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge, where connectivity is achieved solely through shared edges (4-connected adjacency in the square grid), excluding diagonal or corner contacts.^[2] This construction ensures the figure is simply connected without holes, embedded in the infinite square lattice.^[3] The basic unit is the monomino, consisting of a single square. The general term "polyomino" was coined by Solomon W. Golomb in 1953, and formalized in his 1954 paper, extending the naming convention from the domino (two squares) to higher-order figures. Polyominoes of n squares are termed n-ominoes, such as the tromino (n=3), tetromino (n=4), pentomino (n=5), and so on, with the prefix derived from Greek numerals.^[3] For illustration, the two distinct trominoes are the straight tromino (three squares in a row, forming a 1×3 rectangle) and the L-shaped tromino (three squares arranged with two in a row and one attached perpendicularly to one end). These examples highlight the edge-to-edge joining on the grid, where squares align perfectly along their boundaries.^[5] In terms of prerequisite mathematics, polyominoes rely on the topology of the integer lattice \mathbb{Z}^2, where squares are unit cells centered at integer coordinates, and adjacency is defined by the four cardinal directions (up, down, left, right).^[2] This 4-connected graph structure underpins all polyomino constructions, distinguishing them from diagonally connected variants like polyiamonds.

Historical Development

The study of polyominoes traces its origins to early 20th-century recreational puzzles, with the first known pentomino problem appearing in Henry Ernest Dudeney's The Canterbury Puzzles in 1907.^[6] These early puzzles involved arranging connected squares to form larger shapes, laying the groundwork for later systematic exploration in recreational mathematics during the 1950s. The modern field of polyomino research began with Solomon W. Golomb, who introduced the term "polyomino" in a 1953 talk at the Harvard Mathematics Club and formalized it in his 1954 paper "Checker Boards and Polyominoes."^[7] Golomb's work emphasized tiling properties and enumeration, sparking widespread interest. He expanded these ideas in his seminal 1965 book Polyominoes: Puzzles, Patterns, Problems, and Packings, which established foundational concepts and problems in the area.^[3] Martin Gardner further popularized polyominoes through his "Mathematical Games" columns in Scientific American, starting with pentominoes in 1965 and continuing with broader applications in subsequent articles.^[8] Key advancements in tiling theory came from John H. Conway, who, collaborating with Jeffrey C. Lagarias, developed combinatorial group-theoretic methods to determine whether a polyomino tiles the plane in their 1990 paper "Tiling with Polyominoes and Combinatorial Group Theory."^[9] In the 1970s, computational approaches advanced enumeration, with David A. Klarner contributing algorithms and bounds for counting polyominoes up to higher orders.^[2] Recent progress includes high-speed computations using transfer-matrix methods, enabling enumeration of fixed polyominoes up to size 59 in 2025.^[10] Polyominoes have profoundly influenced recreational mathematics and puzzle design, inspiring games like Tetris, which uses tetrominoes as falling pieces.^[11] Their study continues to bridge combinatorics, geometry, and computing, fostering innovations in both academic research and popular entertainment.

Enumeration

Types of Polyominoes

Polyominoes are categorized into three primary types based on the treatment of rotations and reflections under the symmetries of the square grid: fixed, one-sided, and free polyominoes.^[1] Fixed polyominoes count every distinct orientation and reflection as separate, without allowing any transformations; for instance, the domino yields two fixed polyominoes, one horizontal and one vertical.^[12] One-sided polyominoes treat reflections (flips) as distinct but permit rotations, so chiral pairs like mirror images that cannot be superimposed by rotation alone are counted separately; this results in seven one-sided tetrominoes. Free polyominoes consider both rotations and reflections as equivalent, merging chiral pairs into single entries; thus, there are five free tetrominoes, conventionally named the straight (I), square (O), T, L (encompassing its mirror J), and skew (S, encompassing its mirror Z).^[13] The enumeration counts for these types grow rapidly with the number of squares n, reflecting the increasing complexity of connections. The following table summarizes the numbers for small n up to hexominoes:

n (cells)	Free	One-sided	Fixed
1 (monomino)	1	1	1
2 (domino)	1	1	2
3 (tromino)	2	2	6
4 (tetromino)	5	7	19
5 (pentomino)	12	18	63
6 (hexomino)	35	60	216

These values are established through exhaustive enumeration and correspond to OEIS sequences A000105 (free), A000988 (one-sided), and A001168 (fixed).^[14]^[12]

Symmetry and Equivalence

In polyomino theory, symmetries are analyzed using the dihedral group D_4, which consists of eight elements representing the rotational and reflectional transformations that preserve the square grid: rotations by $0^\circ, $90^\circ, $180^\circ, and $270^\circ, along with reflections over the horizontal axis, vertical axis, and two diagonals.^[3] These symmetries form the group under composition, acting on the set of all possible polyomino configurations to determine equivalence classes.^[15] Two polyominoes are considered equivalent under these transformations if one can be obtained from the other by a sequence of rotations and reflections in D_4, disregarding translations and translations combined with these operations. This defines the free polyominoes, where mirror images are treated as identical; in contrast, one-sided polyominoes consider only rotations (the cyclic subgroup C_4) and treat reflections as distinct, while fixed polyominoes are inequivalent unless identical without any rotation or reflection.^[3] To count the number of distinct free polyominoes of a given order, Burnside's lemma is applied, which states that the number of orbits (equivalence classes) is the average number of fixed points over all group elements: \frac{1}{|G|} \sum_{g \in G} \fix(g), where G = D_4 and \fix(g) is the number of polyominoes invariant under symmetry g.^[16] For instance, the identity element fixes all polyominoes, while a $90^\circ rotation fixes only those with fourfold rotational symmetry, and reflections fix those symmetric across the respective axis. This averaging accounts for overcounting due to symmetries, enabling enumeration of free polyominoes from fixed ones.^[3] A representative example is the L-tetromino, formed by three squares in a row with one attached to an end; it has eight distinct orientations under D_4 (four rotations, each with a mirror image), but Burnside's lemma yields one free equivalence class since all are intertransformable. In contrast, the O-tetromino, the 2x2 square, is fixed by all eight group elements due to its full symmetry, resulting in exactly one free form and illustrating how high symmetry reduces the number of distinct representatives.^[16]

Counting Algorithms

One of the foundational computational approaches for enumerating fixed polyominoes is the inductive algorithm developed by D. Hugh Redelmeier in 1981, which employs backtracking to construct polyominoes incrementally by adding cells one at a time while ensuring connectivity.^[17] This method avoids generating duplicates by enforcing a canonical ordering, such as building only in directions that maintain a lexicographically minimal representation of the polyomino's bounding box, thereby systematically exploring the space of all possible connected shapes without redundancy.^[17] Redelmeier's algorithm extended prior enumerations significantly, computing fixed polyomino counts up to n=24 at the time, and it remains a benchmark for generating polyominoes cell-by-cell due to its simplicity and adaptability for parallelization.^[17] To enumerate free polyominoes from fixed counts, computational implementations of Redelmeier's method often integrate group theory via Burnside's lemma, averaging over the symmetries of the dihedral group D_4 (rotations and reflections) to account for equivalences, though this adds overhead for larger n.^[18] For instance, software tools like those implementing Redelmeier's backtracking with symmetry pruning can generate and classify free polyominoes by normalizing shapes to their fundamental domains under group actions.^[2] The transfer-matrix method provides an alternative for counting fixed polyominoes, particularly efficient for large n, by modeling the growth of polyominoes row-by-row on a lattice strip. In this approach, states represent the configuration of occupied cells and connections in the current row, with transitions defined by a matrix M that encodes valid additions of the next row while preserving connectivity across the entire structure. The number of fixed n-ominoes A_n spanning a strip of fixed width can then be computed via the recurrence involving matrix powers, such as A_n = \sum_{states} (M^{n-1})_{s,s'} where the sum is over initial and final states s and s', allowing dynamic programming to build counts iteratively without explicit enumeration of all shapes. Iwan Jensen's 2003 refinement of this method optimized state pruning and matrix sparsity, enabling faster computations by reducing the effective state space from exponential to more manageable sizes for strip widths up to around 10-15 cells.^[2] Recent computational records for free polyominoes have pushed enumerations to n=59 using hybrid approaches combining transfer-matrix methods with multi-threading and symmetry exploitation, as reported in 2025 numerical computations that verified prior counts up to n=50 and extended them, yielding 550,453,451,340,976,338,599,795,923,539,996 free 59-ominoes.^[10] Fixed polyominoes have been enumerated up to n=70.^[12] For free polyominoes, similar techniques, including parallel backtracking variants of Redelmeier's algorithm, have achieved counts up to n=59.^[10] Tools such as custom implementations in languages like C++ or Maple, often building on Jensen's transfer-matrix framework, facilitate these records by distributing state explorations across clusters. Enumerating polyominoes faces inherent challenges due to the exponential growth in their numbers—roughly 4.06^n for fixed polyominoes—necessitating optimizations like early symmetry reduction during generation to prune equivalent branches and cut down on redundant computations.^[10] For free polyominoes, incorporating rotational and reflectional checks mid-generation further mitigates the combinatorial explosion, though storage and verification of canonical forms remain bottlenecks for n beyond 60.^[18]

Asymptotic Enumeration

The asymptotic enumeration of polyominoes focuses on the exponential growth rate of their counts as the size n becomes large, providing approximations rather than exact values. For fixed polyominoes, which are distinguished up to translation only and denoted by a_n, the number satisfies

a_n \sim \frac{c \mu^n}{n}

as n \to \infty, where \mu \approx 4.06257 is the growth constant (also known as Klarner's constant) and c \approx 0.3169 is the amplitude constant.^[19] This form arises from the dominant singularity in the generating function A(u) = \sum a_n u^n, analyzed via transfer-matrix methods that model polyomino growth by tracking boundary configurations on a cylinder, combined with singularity analysis using differential approximants on series data.^[19] The value of \mu is the reciprocal of the critical fugacity u_c \approx 0.24615, corresponding to the largest eigenvalue of the transfer matrix in the limit of infinite width.^[2] The growth constant \mu has been bounded rigorously: a lower bound of \mu > 4.00253 was established using enumerations on twisted cylinders and finite-state automata to model connectivity, while upper bounds have improved to \mu \leq 4.5238 via recurrence-based methods for irreducible polyominoes (twigs) with dead-end cells.^[20]^[21] These bounds confirm the existence of \mu = \lim_{n \to \infty} [a_{n+1}/a_n], first proved for lattice animals in general.^[22] For free polyominoes, counted up to rotations and reflections and denoted by f_n, the asymptotic form is similar, f_n \sim \frac{c' \mu^n}{n}, with the same growth constant \mu \approx 4.06257 but adjusted amplitude c' \approx c/8 \approx 0.0396, since nearly all large polyominoes lack nontrivial symmetries under the dihedral group D_4 of order 8.^[19] The exact count f_n is obtained via Burnside's lemma as the average number of fixed polyominoes over the 8 group elements: f_n = \frac{1}{8} \sum_{g \in D_4} \fix(g), where \fix(g) is the number invariant under symmetry g; the subleading terms account for the vanishing density of symmetric cases.^[2] Empirical fits to the asymptotic form have been refined using extensive computations: early work enumerated fixed polyominoes up to n=24 and free up to n=18, enabling initial estimates of \mu, while parallel transfer-matrix implementations extended fixed counts to n=56 with error bounds on \mu of \pm 5 \times 10^{-7}. More recent advances, building on improved algorithms, have reached n=59 for free polyominoes and n=70 for fixed, yielding fits with relative errors below 0.1% for large n and tighter bounds like \mu = 4.0625696 \pm 0.0000005. Key contributions span from Klarner's introduction of \mu in the 1960s, through Conway's algorithmic foundations in the 1970s, to Jensen's series analyses in the 2000s and Barequet's high-performance enumerations in the 2020s.^[2]^[19] Perimeter-area relations connect to these asymptotics through refined generating functions that track boundary length alongside area n; the growth constant \mu emerges from the radius of convergence in the perimeter-weighted series, where adding a cell typically alters the perimeter by amounts reflecting local boundary states in the transfer matrix, linking exponential growth to average boundary scaling as O(\sqrt{n}) for compact shapes.^[19]

Special Polyomino Classes

Straight polyominoes, also known as I-n-ominoes, consist of n unit squares aligned in a single row or column, forming a linear chain. These are the simplest subclass of polyominoes, with exactly one free I-n-omino for each n ≥ 1, as horizontal and vertical orientations are equivalent under rotation. In fixed enumerations, there are two distinct I-n-ominoes for n ≥ 2 (horizontal and vertical), while one-sided counts coincide with the free count due to reflection symmetry.^[23] Skew polyominoes, commonly referred to as Z-polyominoes, feature zigzag patterns where squares alternate in direction, exemplified by the Z-tetromino for n=4. This class generalizes to longer zigzags, maintaining connectivity without sharp 90-degree turns in certain directions. Enumeration of skew polyominoes relies on recurrence relations derived from generating tree decompositions or transfer-matrix methods, allowing systematic counting by area or perimeter. A related subclass, Z-convex polyominoes—where any two cells are joined by a monotone path with at most two direction changes—admits a closed-form generating function P(t) = \sum p_n t^{n+2}, with coefficients yielding the number by semi-perimeter; for instance, the series begins t^2 + 2t^3 + 7t^4 + 28t^5 + 116t^6 + \cdots, corresponding to small-area counts starting from the domino. For hexominoes (n=6), there is one free skew hexomino, the extended Z-shape, though broader Z-convex variants number 116 by the generating function metric.^[24]^[25] Tree-like polyominoes, including polyplets as branching variants, are characterized by a dual adjacency graph that forms a tree, ensuring no cycles and thus simply connected structures without loops. These allow branching, as in the V-tromino (a corner-connected or edge-adjacent Y-shape), distinguishing them from linear forms. Polyplets extend this by permitting connections at edges or corners, enabling more compact branching; their enumeration counts fixed n-polyplets (with holes allowed) as 1, 2, 5, 22, 94, 524 for n=1 to 6. Tree-like polyominoes (edge-connected only) have been enumerated via bijections and recursive formulas for saturated cases, where all leaves are fully extended; for n=6, the number of fixed tree-like hexominoes aligns with broader counts in small-n limits, though cycles exclude shapes like the O-tetromino. Detailed bijections map saturated tree-like hexominoes of size 25 (5×5 envelope) to smaller sets, highlighting structural constraints.^[26]^[27]^[28] Directed polyominoes, or directed lattice animals, form a subclass where cells are added recursively only above or to the right of existing cells, restricting growth to positive directions and excluding holes by construction. Lattice animals generally denote simply connected polyominoes without holes, contrasting with general polyominoes that may contain cycles or enclosed voids; directed variants further impose acyclicity in growth paths. The enumeration follows the generating function D(z) = \frac{1}{2} \left( \frac{\sqrt{1+z}}{1-3z} - 1 \right), yielding fixed counts via series expansion, with small values d(1)=1, d(2)=2, d(3)=5, d(4)=14, d(5)=42, d(6)=132. For hexominoes, this gives 132 fixed directed forms, fewer than the total 216 fixed hexominoes due to directional constraints; free counts reduce further under symmetry. Simply connected lattice animals without directionality match total polyomino counts for n≤8 (all 35 free hexominoes are hole-free), with enumerations extending to n=42 via computational methods.^[2]^[29]

Tiling Properties

Tiling Finite Regions

Polyominoes are often used to tile finite regions, such as rectangles or other bounded shapes, by placing multiple copies or complete sets without gaps or overlaps. A prominent example involves the twelve distinct pentominoes, each covering five unit squares, which together cover exactly 60 squares and can tile rectangles of dimensions 3×20, 4×15, 5×12, or 6×10. These tilings demonstrate how a fixed set of polyominoes can exactly cover rectangular regions whose area matches the total area of the set, with 2 solutions for 3×20, 368 for 4×15, 1010 for 5×12, and 2339 for 6×10.^[30]^[31] Solomon Golomb introduced concepts of deficiency in polyomino tilings, where a region has fewer squares than a multiple of the polyomino's size. For L-trominoes (three-square polyominoes), Golomb proved that a deficient 2^n × 2^n square (missing one square) can always be tiled, using a recursive divide-and-conquer approach that places an L-tromino at the center to handle the deficiency and tiles the remaining quadrants inductively.^[32] This theorem extends to more general deficient rectangles under conditions like total area congruent to the deficiency modulo the polyomino size, highlighting how polyominoes can adapt to imperfect regions while maintaining coverage.^[33] When replicating a single polyomino type to tile a rectangle, basic conditions must hold: the rectangle's area must be divisible by the polyomino's size, ensuring an integer number of copies. For tetrominoes (four-square polyominoes), all five free tetrominoes can tile rectangles, but additional parity constraints arise from checkerboard colorings, where each tetromino covers two black and two white squares, requiring the rectangle to have equal black and white squares. Specific tetrominoes impose further restrictions; for instance, the T-tetromino requires both dimensions divisible by 4 for rectangular tilings.^[34] Impossibility results often rely on invariant arguments like coloring. The classic mutilated chessboard problem illustrates this: an 8×8 chessboard with two opposite corners removed (both the same color, leaving 32 squares of one color and 30 of the other) cannot be tiled by 32 dominoes (each covering one black and one white square), as the color imbalance prevents exact coverage. Solving polyomino tiling problems for finite regions computationally frequently reduces to the exact cover problem, where the goal is to select placements that cover each square exactly once. Donald Knuth's Dancing Links (DLX) algorithm efficiently implements backtracking for this, using a doubly-linked matrix to prune search spaces rapidly; it has been applied to enumerate all 2339 solutions for the 6×10 pentomino tiling.^[35] This method scales well for puzzle instances up to moderate sizes, enabling exhaustive exploration of feasible tilings.

Tiling the Infinite Plane

Polyominoes can tile the infinite plane in periodic arrangements, where the tiling repeats according to a lattice structure generated by translations, rotations, or reflections of the prototile. A key tool for determining such tilings is the Conway criterion, which provides a sufficient condition for a polyomino to tile the plane using these symmetries: the boundary of the polyomino must be divisible into six arcs such that opposite arcs match under translation, and the remaining arcs satisfy rotational or reflectional symmetry. This criterion, while not necessary, successfully identifies all known tileable polyominoes up to order 8, either individually or in pairs.^[36] Certain polyominoes, known as rep-tiles, enable periodic tilings through hierarchical dissection, where the plane is covered by repeatedly subdividing larger copies into smaller congruent versions of the same shape. The L-tromino exemplifies a rep-4 tile, as it can be divided into four smaller L-trominoes scaled by a factor of 1/2, allowing recursive construction of a periodic tiling across the plane.^[3] Introduced by Solomon Golomb, rep-tiles among polyominoes often exhibit self-similar properties that facilitate lattice-based coverings. For single polyomino tilings of the plane, all five free tetrominoes are known to admit periodic tilings, typically by translation or with minimal rotations.^[37] All monominoes through hexominoes tile the plane, while among heptaminoes, only four of the 108 fail to do so; similar classifications hold for octominoes. However, for larger orders, determining tileability remains open for some polyominoes due to computational complexity, though the general decidability of the mono-polyomino tiling problem is unresolved. Aperiodic tilings of the plane using polyominoes involve sets that admit coverings but forbid periodic arrangements. Such sets can be constructed by encoding aperiodic Wang tiles—colored unit squares with matching rules—into polyominoes via protrusions and indentations that enforce adjacency constraints.^[38] For instance, sets as small as three polyominoes exist that force only aperiodic tilings, mirroring the undecidability of the general polyomino tiling problem. Heesch's problem addresses non-tiling polyominoes by quantifying their "corona number," the maximum number of concentric layers of congruent copies that can surround a central instance without gaps or overlaps. Polyominoes with finite Heesch numbers, such as certain 11-ominoes with Heesch number 3, illustrate barriers to full plane tilings, with the highest known for polyominoes being 6 (as of 2021).^[39]^[40] This measure connects to aperiodic phenomena by highlighting shapes that extend locally but resist global periodicity.^[41]

Notable Tiling Theorems

One of the foundational results in polyomino tiling theory is Golomb's theorem, which establishes a key implication for rectifiability. Specifically, if a polyomino tiles some rectangle, then it also tiles a larger scaled-up copy of itself. This result forms part of Golomb's broader hierarchy of tiling capabilities, where rectifiability implies successively weaker conditions, such as tiling half-strips or quadrants, down to self-replication at larger scales. The converse—whether every polyomino that tiles a larger copy of itself is rectifiable—remains an open problem.^[42] Golomb's proof relies on constructing iterative enlargements using the rectangular tiling as a base, demonstrating how the polyomino can replicate its boundary structure in scaled versions. Coloring arguments provide simple yet powerful obstructions to polyomino tilings, particularly for finite regions like mutilated boards. For dominoes, which are the simplest polyominoes covering two squares, a standard 2-coloring of the grid (e.g., black and white like a chessboard) shows that each domino covers one square of each color. Thus, any region tilable by dominoes must have an equal number of black and white squares. The classic mutilated chessboard problem illustrates this: removing two opposite corners from an 8×8 chessboard eliminates two squares of the same color, leaving 32 of one color and 30 of the other, making domino tiling impossible. This argument generalizes to higher-order colorings for larger polyominoes; for example, a 3-coloring can obstruct tromino tilings of deficient rectangles where color imbalances occur. Such invariants are computable and often suffice to prove non-tilability without exhaustive search. The undecidability of polyomino tilings traces back to Berger's seminal work on the domino problem, which showed that determining whether a given set of tiles can cover the infinite plane is algorithmically unsolvable. Berger's proof reduces the halting problem for Turing machines to tiling with Wang tiles (colored squares with matching rules), establishing undecidability for periodic tilings and extending to aperiodic cases. This result has been adapted to polyominoes: given a finite set of polyominoes, deciding if they can tile the plane (allowing translations and rotations) is undecidable, with reductions showing the threshold at as few as 5 polyominoes (as of 2025).^[43] Implications include that no general algorithm exists to classify tile sets by plane-tiling capability, impacting computational approaches to tiling problems. For polyominoes with holes, tiling obstructions can arise from invariants analogous to the Dehn invariant in higher dimensions, which measures angular defects and prevents certain dissections or space-filling despite matching areas. Although standard polyominoes are simply connected without holes, holed variants may fail to tile regions or the plane if their internal topology introduces mismatches in boundary curvature or genus that cannot be resolved by congruent copies. This extends Dehn's ideas from 3D polycubes, where nonzero Dehn invariants block Euclidean space tilings, to 2D cases where holed polyominoes exhibit similar barriers to complete coverage.

Applications in Puzzles and Games

Classic Polyomino Puzzles

Pentomino puzzles represent some of the most enduring challenges in polyomino assembly, requiring the 12 distinct pentomino pieces—each covering five unit squares—to fill rectangular regions without gaps or overlaps. A notable example is the 3×20 rectangle tiling, which tasks solvers with arranging these 12 pieces into a narrow 3×20 rectangle; due to the constrained width, only 2 unique solutions exist, emphasizing precise interlocking and spatial reasoning.^[44] In comparison, assembling the same set into a 6×10 rectangle yields 2339 solutions, offering greater variety while still demanding systematic trial and error to discover valid configurations.^[44] These rectangle tilings highlight the balance between puzzle difficulty and solution multiplicity, with the 3×20 variant prized for its rarity and the 6×10 for its abundance. Tetromino tangrams adapt the principles of shape dissection using the five free tetrominoes, which together cover 20 unit squares, to form intricate silhouettes such as animals, letters, or other figures. Solvers rotate and translate these pieces— the straight I, square O, T, L/J (mirror pair), and S/Z (skew pair)—to replicate outlined targets, fostering creativity akin to traditional tangram play but with standardized polyomino forms.^[45] This variant encourages free-form assembly beyond strict rectangles, often resulting in dozens of possible interpretations per target shape, and serves as an accessible entry point to polyomino recreation. The Soma cube extends polyomino dissections into three dimensions, employing seven polycubes—irregular solids derived from combining triominoes and tetrominoes extruded to varying heights—to construct a 3×3×3 cube. Rooted in two-dimensional assembly logic, this puzzle has exactly 240 distinct solutions when disregarding rotations and reflections, showcasing how polyomino concepts scale to volumetric challenges.^[46] Its design promotes layered building, where base polyomino-like footprints must align across levels for stability. Early historical polyomino puzzles drew from domino assembly variants explored in the early 20th century, evolving into structured challenges that influenced modern designs. Contemporary tools, such as the Polyform Puzzler software and the Polyomino Solver Pro mobile app, automate solution enumeration for pentomino rectangles and tetromino dissections, enabling users to verify configurations or explore thousands of assemblies efficiently.^[45]^[47] These digital solvers, often employing backtracking algorithms, democratize complex puzzle-solving while preserving the intellectual satisfaction of manual attempts.

Polyominoes in Board Games

Polyominoes have found prominent use in board games, where they serve as key components for strategic placement and spatial competition. One of the most notable examples is Blokus, invented by French mathematician Bernard Tavitian and first published in 2000 by Sekkoïa.^[48] In this abstract strategy game for 2 to 4 players, each participant receives a set of 21 polyominoes ranging from monominoes to pentominoes in four colors, which they place on a 20x20 grid.^[49] Placement rules require that the first piece occupy a corner of the board and subsequent pieces touch at least one corner of the same color while not sharing edges, promoting careful spatial planning to maximize coverage while hindering opponents.^[49] Tetris, originally developed by Alexey Pajitnov in 1984 as an electronic game but adapted into various board game and digital formats, centers on tetrominoes—the seven one-sided tetromino shapes that include the straight, square, T, L, J, S, and Z forms. Players manipulate falling tetrominoes by rotating and positioning them to fill horizontal lines on a grid, clearing completed lines for points and preventing stack overflow.^[50] Scoring varies by version but typically rewards line clears, with bonuses for simultaneous multi-line completions (e.g., a "Tetris" for four lines yields higher points), emphasizing efficient rotation and placement under time pressure.^[50] Polyominoes also appear in variants extending traditional domino games, which employ diominoes as the basic polyomino unit. Mexican Train, a popular multiplayer adaptation of dominoes using a double-12 set, involves players building chains from a central hub while managing open "trains" to score low by minimizing unmatched ends.^[51] This game highlights polyomino chaining mechanics, where strategic branching and blocking force opponents into high-scoring positions. Digital implementations of polyomino-based games, such as adaptations of Tetris, further explore these elements in competitive or solo formats. Beyond these classics, polyominoes continue to inspire modern board games as of 2025, including titles like Patchwork (2009), Isle of Cats (2019), and Foundations of Rome (2020), which incorporate polyomino placement for resource management and spatial strategy.^[52]^[53]^[54] Strategic depth in polyomino board games often revolves around blocking opponents and controlling board space to limit their options while optimizing one's own placements. In Blokus, for instance, players aim to encroach on central areas, denying rivals room for larger pieces. Expansions and variants, such as Blokus Duo or Trigon, introduce modified rules or larger polyomino sets to enhance replayability and tactical complexity.^[49]

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A001168 - OEIS
The currently best-known lower and upper bounds on this constant are 3.9801 (Barequet et al., 2006) and 4.6496 (Klarner and Rivest, 1973), respectively. But see ...
[12]
Tetromino -- from Wolfram MathWorld
A tetromino is a 4-polyomino. There are five free tetrominoes, seven one-sided tetrominoes, and 19 fixed tetrominoes.
[13]
A000105 - OEIS
The number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes.Missing: fixed | Show results with:fixed
[14]
Enumeration of Symmetry Classes of Convex Polyominoes in the ...
By virtue of Burnside's lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of 4and 4.
[15]
Counting polyominoes: Yet another attack - ScienceDirect.com
A polyomino enumeration method, faster than any previous, is presented. This method includes the calculation of the number of symmetric polyominoes.
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Counting Polyominoes, Revisited
[Red81] D.H. Redelmeier. Counting polyominoes: yet an- other attack. Discrete Mathematics, 36(2):191–203,. 1981. [S+18] N.J.A. Sloane et al. The on-line ...
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[PDF] Statistics of lattice animals (polyominoes) and polygons - arXiv
Abstract. We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46.
[18]
[PDF] λ > 4 - Freie Universität Berlin
Aug 19, 2015 · We investigated a fundamental question related to polyominoes, namely, what is their growth constant, the asymptotic ratio between A(n + 1) and ...<|control11|><|separator|>
[19]
[PDF] Improved Upper Bounds on the Growth Constants of Polyominoes ...
Jun 27, 2019 · Since then, λ2 has been called “Klarner's constant.” Only in 1999, Madras [22] proved the existence of the asymptotic growth rate, namely, limn ...
[20]
[PDF] De Bruijn Polyominoes - arXiv
May 28, 2024 · A straight polyomino is a polyomino with at least 2 cells whose cells all oc- cupy a single row of the square lattice.1 Straight polyominoes of ...
[21]
[PDF] The number of Z-convex polyominoes - garsia at york
Abstract. In this paper we consider a restricted class of polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any ...
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[PDF] Generation of Skew Convex Polyominoes - CEUR-WS.org
Sep 13, 2024 · Skew polyominoes are generated using a recursive method based on generating trees, with local expansions, and a CAT algorithm for exhaustive ...Missing: recurrence | Show results with:recurrence
[23]
Saturated fully leafed tree-like polyforms and polycubes
In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4 k + 1 and the set of tree-like polyominoes of size k.Missing: polyplets variants
[24]
Polyplet -- from Wolfram MathWorld
A polyomino-like object made by attaching squares joined either at sides or corners. Because neighboring squares can be in relation to one another as kings ...
[25]
[PDF] Saturated Fully Leafed Tree-Like Polyforms and Polycubes - arXiv
Mar 24, 2018 · In the polyomino case, we present a bijection between the set of saturated tree-like polyominoes of size 4k + 1 and the set of tree-like ...
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[PDF] Hole-free Partially Directed Animals | DLT 2019
The approach can be extended to partially directed animals with a fixed number of holes. Page 3. introduction. Exhaustive generation. Polyominoes.Missing: excluding | Show results with:excluding
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Equivalence Classes Among Pentomino Tilings of the 6x10 Rectangle
Jan 10, 1991 · 1. The problem. Many workers have found the 2339 unique solutions to tiling the 6 x 10 rectangle with pentominoes [Golomb, 1965]. (Pentomino is ...
[28]
Tiling Deficient Rectangles with Trominoes - jstor
Trominoes were introduced by Golomb [3], who proved that deficient squares whose side length is a power of two can be tiled. Chu and Johnsonbaugh first extended.
[29]
[PDF] Tiling rectangles and deficient rectangles with trominoes
The first three rectangles are tilable by Theorem 1, while the last term is a RR x RR square of deficiency 1 that is tilable by the original theorem of Golomb ...
[30]
[PDF] Tilings of rectangles with T-tetrominoes - UCLA Mathematics
Aug 26, 2003 · Theorem 2 (Walkup) If an m × n rectangle can be tiled by T-tetrominoes, then both m and n must be divisible by 4. Furthermore, all segments ...
[31]
Tiling a Rectangle with L-tetrominoes
Find a necessary and sufficient condition that an a×b rectangle can be exactly covered (completely, and without overlaps) with L-tetrominoes.
[32]
[cs/0011047] Dancing links - arXiv
Nov 15, 2000 · The second trick is to add a ghost square that represents the identity of each polyomino. Thus puts the rule that each polyomino be used once on ...Missing: exact | Show results with:exact
[33]
Tiling the plane - The Labyrinth of Polyominoes
A polyomino tiles the plane by translation if and only if it satisfies the translation criterion. The Conway Criterion. The translation criterion covers some ...Missing: periodic aperiodic rep- Heesch
[34]
[PDF] Isohedral Polyomino Tiling of the Plane - People - University of Florida
Penrose [11] has constructed aperiodic sets of tiles, some of which have as few as two elements. For example, there are uncountably many tilings of the plane by ...
[35]
[PDF] Tiling the Plane with a Fixed Number of Polyominoes
Tile sets with a biperiodic tiling are recursively enumerable. Undecidability is to be found in aperiodic tile sets, tile sets that only admit aperiodic tilings ...
[36]
[PDF] Heesch Numbers of Unmarked Polyforms
Because polyominoes, polyhexes, and polyiamonds are subsets of ambient regular tilings of the plane, it is possible to reduce the geometric problem of ...
[37]
https://people.clas.ufl.edu/avince/files/Isohedral.pdf
[38]
Tiling with polyominos
here's the diagram of implications he gives: rectangle / | / | / v / / half strip / / | / | / v / \/_ bent strip itself | | \ v \ \ quadrant & strip \ \ / \ \ / ...
[39]
Pentominoes: Puzzles & Solutions - Polyform Puzzler
Misc. 3x20 ring (3x20 rectangle joined at the ends): 2 solutions (same as the 3x20 rectangle; the V piece forces a partition).Missing: burr | Show results with:burr
[40]
Polyform Puzzler
Polyform Puzzler is a set of solvers for many polyform puzzles (like Pentominoes and Soma Cubes), and a software toolkit for exploring & solving polyform ...Missing: enumeration | Show results with:enumeration
[41]
Soma Cube : 3 Steps - Instructables
There are 240 distinct solutions of the Soma cube puzzle, excluding rotations and reflections: these are easily generated by a simple recursive backtracking ...
[42]
https://www.ics.uci.edu/~eppstein/junkyard/polyom-tile.html
[43]
Board Game Step Ladder – Polyominos - Meeple Mountain
Jan 28, 2019 · Blokus. Invented by Bernard Tavitian and first published in 2000, Blokus has 2 to 4 players vying to place more of their polyominoes onto the ...
[44]
Blokus | Board Game - BoardGameGeek
In stock Rating 3.4 (28,968) Blokus (officially pronounced "Block us") is an abstract strategy game with transparent, Tetris-shaped, colored pieces that players are trying to play onto ...Missing: invention | Show results with:invention
[45]
Scoring - TetrisWiki
Oct 15, 2025 · Most games award points to the player for completing various tasks. The earliest games awarded points only for dropping tetrominoes, ...
[46]
Mexican Train Dominoes Online: Play Free Game with Official Rules
Play, practice, strategize, and have fun against computer-generated players in this free online version of Mexican Train Dominoes.Mexican Train rules · Clarifying Mexican Train... · Free online Mexican Train...