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Connection game

A connection game is an abstract strategy game for two players in which the objective is to connect designated parts of the board, such as opposite sides or corners, by placing pieces on a grid. These impartial games, often played on hexagonal or other symmetric boards, require players to alternately claim cells while blocking the opponent, with the first to achieve a connecting chain winning. Common board types include hex-hexagonal grids, and rules may include variations like multiple connection goals or capturing.^[1] The genre originated with Hex, invented by Danish mathematician and poet Piet Hein in 1942 and independently discovered by American mathematician John Nash around 1948.^[2] Since then, numerous variants have been developed, emphasizing mathematical strategy and perfect information.^[1]

Definition and Fundamentals

Core Characteristics

Connection games constitute a genre of abstract strategy board games played by two impartial players who alternate turns placing undifferentiated pieces on a board to form topological connections, typically unbroken paths or contiguous groups linking designated points such as opposite edges.^[3]^[4] These games exhibit core traits of abstract strategy, including the absence of narrative themes, perfect information available to both players, and no random elements, with outcomes determined solely by sequential decision-making focused on spatial positioning and mutual blocking.^[5] The emphasis lies on strategic control of board space to advance one's connectivity while obstructing the opponent's.^[4] Boards in connection games commonly employ hexagonal grids for their natural six-directional connectivity, square grids for orthogonal or diagonal links, or irregular shapes tailored to specific connection goals, prioritizing the establishment of continuous networks over mechanisms like piece capture.^[6]^[1] This genre differs from territory-based abstracts like Go, where players enclose areas and capture stones to score points, by centering victory on direct linkage without territorial accumulation or removal of pieces.^[7] It also contrasts with alignment games such as tic-tac-toe, which demand brief linear sequences, whereas connection games typically require extended, board-spanning paths that demand deeper tactical depth.^[3]

Basic Rules and Objectives

Connection games are abstract strategy board games in which two players alternate turns to place their pieces on an empty board, typically consisting of a grid of cells or intersections, with the aim of forming a specific type of connection using their pieces.^[4] Players use unmarked pieces, often distinguished by color such as black and white, and place one piece per turn on unoccupied positions, following impartial rules that apply equally to both sides.^[4] This turn-based structure ensures symmetric gameplay, where neither player has an inherent advantage from the rules themselves, though first-player benefits may arise due to the finite board space.^[4] The primary objective in most connection games is to create a continuous path or group of one's own pieces that links two or more designated goals on the board, such as opposite edges or corners, before the opponent achieves the same.^[4] Connections are formed by adjacency, where pieces touching along edges (in hexagonal or square grids) or at vertices count as linked, preventing the opponent's pieces from blocking all possible paths.^[4] For example, in the archetypal game of Hex, players win by connecting their pair of opposite board sides with an unbroken chain of pieces.^[8] Secondary objectives appear in variants, such as forming closed loops that enclose territory or bridging multiple non-adjacent points, though these build on the core path-forming goal.^[4] Draws are rare due to topological properties ensuring one player must connect first in edge-to-edge formats, but some games allow for alternative win conditions like capturing pieces or achieving multiple connections simultaneously.^[4] Conceptually, connection games can be modeled as graphs where the board represents vertices or edges that players claim through piece placement, with victory determined by the existence of a path in the induced subgraph of one's pieces.^[4] This notional space highlights the combinatorial nature of the games, emphasizing strategic control of critical junctions to secure connectivity.^[4]

Historical Development

Early Origins

The earliest known connection game is Lightning, patented in 1892 by Harry A. Doty of New Haven, Connecticut.^[9] This board game involves players placing pieces with line segments on a grid to form a continuous path from one side to the opposite, marking it as the first documented example of connectivity-focused play.^[9] Doty's invention emphasized strategic line-building on a subdivided board, with players alternating turns to connect while potentially blocking opponents, transitioning recreational sketching into structured competition.^[9] In the late 19th and early 20th centuries, precursors like Zig-Zag, published by Parker Brothers in 1932, built on this foundation by introducing piece-based mechanics for chain formation across a board. These early alignment games shifted focus toward pure connectivity, influencing later designs such as Hex. Other informal variants emerged anonymously through parlor amusements, reflecting a growing interest in abstract strategy without thematic elements. Connection games arose primarily in Europe and North America amid the 19th-century surge in abstract puzzles, such as checkers variants and geometric challenges, which popularized non-narrative board play. Unlike ancient territory games like Go—dating back over 2,500 years in East Asia—connection games lack pre-modern analogs, as historical records show no equivalent emphasis on linear paths or side-to-side linkages in antiquity. This emergence aligned with industrialization's promotion of portable, intellectual recreations for urban households. Key figures in these origins were largely anonymous inventors of recreational sketches, with Doty as a notable exception for formalizing Lightning via patent. The period marked a shift from ad hoc paper games to commercialized versions, laying groundwork for connectivity as a core mechanic in abstract strategy.

Modern Innovations

The modern era of connection games began with the invention of Hex in 1942 by Danish mathematician and poet Piet Hein, who initially called it "Polygon" and described it in the Danish newspaper Politiken.^[10] The game was independently rediscovered in 1948 by American mathematician John Nash while he was a graduate student at Princeton University, though some accounts question the extent of his independent development.^[10] Hex gained widespread attention in the United States through Martin Gardner's July 1957 column in Scientific American, which popularized it among recreational mathematicians and hobbyists.^[11] The mid-20th century saw a proliferation of connection games, building on Hex's abstract connection mechanic. Bridg-it, invented by mathematician David Gale around 1958, introduced bridging elements on a grid and was featured in Gardner's October 1958 Scientific American column, highlighting its impartial variant known as the Shannon switching game.^[12] The Game of Y emerged in the early 1950s, first described by mathematicians John Milnor and Claude Shannon, and independently by Craige Schensted in 1953 as "Trictactoe," using a triangular board where players aim to connect all three sides.^[13] TwixT, designed by Alex Randolph in 1962 and published by 3M, added linking pegs on a square grid, emphasizing knight-move connections and becoming a commercial success.^[14] By 1981, Trax, invented by David Smith in 1980 and first published in New Zealand, innovated with portable tiles forming lines on any surface, allowing flexible play without a fixed board.^[15] The 21st century marked a surge in connection game development, aided by digital tools that facilitated prototyping and variant creation. Tak, inspired by the fictional game in Patrick Rothfuss's 2011 novel The Wise Man's Fear, was adapted into a physical board game in 2016 by designer James Ernest through a successful Kickstarter campaign, introducing stacking and placement rules on a square grid.^[16] Software like Ludii, a game design framework, has enabled rapid generation of new connection variants by automating rule testing and board configurations.^[17] Key figures have driven innovation in this period. Cameron Browne, an Australian computer scientist, has designed numerous connection games using AI-assisted methods, as detailed in his 2005 book Connection Games: Variations on a Theme and his PhD work on evolutionary game design, which produced playable variants like Yavalath.^[17] Klaus Winther, through his TwixT Games imprint, published Havannah in the 1990s, bringing Christian Freeling's 1979 hexagonal design to wider audiences and supporting the abstract strategy community.^[18] Combinatorial game theory has provided a formal lens for studying these games, with analyses of impartial variants like Bridg-it using Sprague-Grundy theorem assignments to determine winning strategies, as explored in works by Richard K. Guy and others since the 1980s.^[19] Recent advancements have addressed accessibility gaps through online platforms and AI. Boardspace.net, launched in the early 2000s, hosts multiplayer sessions for dozens of connection games like Hex, Y, and TwixT, fostering global tournaments and community analysis.^[20] Post-2020, AI developments, such as AlphaZero-inspired engines like KataHex, have solved optimal play on larger Hex boards and analyzed game lengths, achieving superhuman performance through self-play reinforcement learning.^[21] Similar AI tools for Tak and other variants have emerged, enhancing training and strategy exploration via neural networks.^[22]

Gameplay Mechanics

Board and Piece Configurations

Connection games typically feature boards constructed from regular tilings, such as hexagonal, rhombus-shaped, or triangular grids, which facilitate the formation of paths through adjacent cells or edges. The rhombus board in Hex, for instance, consists of a hexagonal tiling arranged in an 11x11 configuration for standard play, where each cell has up to six neighbors, balancing accessibility for beginners with strategic depth.^[23] In contrast, Havannah uses a hexagonal board shaped as a larger hexagon with 8 to 10 cells per side, where interior cells have six neighbors and boundary cells fewer, influencing path formation complexity.^[24] The Game of Y employs a triangular board of hexagonal cells, with the standard size being 9 per side as produced by Kadon Enterprises, though larger variants such as 13x13x13 exist, with three equivalent sides to promote symmetric play.^[23]^[25] Board sizes significantly impact game complexity; smaller variants, such as 9x9 hexagonal grids, suit introductory play by reducing the state space, while larger 13x13 or expert-level 10-sided Havannah boards increase computational demands and strategic nuance.^[24] Pieces in most connection games are simple markers, like colored stones placed on cells without movement or removal, as seen in Hex, Y, and Havannah, where players claim unoccupied spaces to build adjacency chains.^[23] Variants introduce more intricate mechanics: Tak utilizes a square grid (e.g., 5x5 standard) with flat stones, capstones, and standing stones (walls), allowing stacking with no height limit, though the number of pieces movable from a stack is limited by the carry rule (board size minus one).^[26] TwixT employs pegs placed on a grid's vertices, connected by links resembling knight moves in chess, with rules permitting link removal in some versions to adjust paths.^[23] From a graph theory perspective, connection game boards are modeled as dual graphs, where cells correspond to vertices and adjacencies to edges, enabling analysis of connectivity between opposite boundaries or corners; in hexagonal tilings like Hex, this yields a planar graph with average degree six, while edge-based games like TwixT operate on the primal graph's vertices and potential edges.^[23] Adaptations extend these foundations, including asymmetric boards that alter side lengths for unbalanced play, three-dimensional grids stacking layers for volumetric paths, or irregular shapes in custom variants to explore non-standard topologies.^[23]

Winning Conditions and Variations

In connection games, the primary winning conditions revolve around establishing a continuous path or linkage between designated board elements, such as opposite edges or specific points. The most common standard win involves creating an edge-to-edge path, where a player connects two opposing sides of the board with an unbroken chain of their pieces, as seen in prototypical examples like Hex.^[1] Another standard form is multi-point linkage, such as connecting three designated corners or sides simultaneously, which generalizes the pairwise connection to require broader control over the board's perimeter.^[13] Less frequently, victory can be achieved through area enclosure, where a player surrounds and claims a region of the board, though this blends connection mechanics with territorial control.^[27] Variants introduce diverse winning criteria to alter strategic depth and board dynamics. For instance, some games allow victory via forming rings (closed loops of pieces) or bridges (links between non-opposite corners), expanding beyond linear paths to include cyclic or angular connections.^[27] In other variants, such as those emphasizing shortest paths, players compete to form the most efficient linking route, or win by blocking the opponent's potential connections entirely, inverting the connective goal into a denial strategy.^[27] Misère versions reverse the standard outcome, where completing a connection results in loss, forcing players to avoid linkage while compelling the opponent to form one; scored play, meanwhile, awards points for multiple partial connections or enclosed areas rather than a single decisive win.^[1] A key principle in many pure connection games is the no-draw theorem, which uses topological arguments to guarantee that exactly one player will achieve victory on a fully occupied board, as the board's structure ensures at least one connecting path must emerge.^[28] This property, proven for games like Hex, relies on the board's finite, connected topology and the impartial placement rules, eliminating stalemates in standard play.^[29] To address first-player advantages inherent in these unbalanced setups, the pie rule is often employed, allowing the second player to swap positions after the initial move, thereby equalizing opportunities.^[30] Rule tweaks further diversify the genre, including placement restrictions that prohibit adjacent pieces or require specific orientations to prevent overcrowding and encourage spatial planning.^[27] Multi-player adaptations extend the two-player format to three or more participants, typically in a free-for-all where each player connects their own designated edges, or in teams with shared linkages, though this can introduce draws if no single victor emerges.^[1] Hybrid variants incorporate capture elements, such as removing opponent's pieces by landing on them during movement, combining connection goals with positional elimination, as in Lines of Action where players maneuver to link all their pieces while capturing to disrupt foes.^[31]

Prominent Connection Games

Hex

Hex is a two-player abstract strategy game played on a rhombus-shaped board composed of 11×11 hexagonal cells, where players alternate placing stones of their color—typically black and white—to form a connected path between their designated opposite sides.^[2] Black aims to connect the left and right edges, while white seeks to connect the top and bottom edges; the first player to achieve such a connection wins.^[2] To address the first-player advantage, the pie rule (also known as the swap rule) is commonly employed: after the first player places one of their stones, the second player may choose to swap colors with the first player or continue normally.^[32] The game was independently invented twice in the mid-20th century. Danish mathematician and poet Piet Hein created it in 1942 while at the Niels Bohr Institute for Theoretical Physics in Copenhagen, initially calling it Polygon and introducing it as an intellectual diversion among colleagues.^[2] American mathematician John Nash rediscovered it around 1948 during his time at Princeton University, where it gained traction among game theory enthusiasts who referred to it as Nash.^[10] Interest surged in 1957 when Martin Gardner featured Hex in his "Mathematical Games" column in Scientific American, describing its rules and topological properties, which introduced the game to a broader audience and solidified its place in recreational mathematics.^[11] A defining feature of Hex is that draws are impossible, a property proven using topological arguments equivalent to the Brouwer fixed-point theorem: in any complete filling of the board, one player's stones must connect their opposite sides due to the board's continuous boundary structure.^[33] Determining the winner from a given position is computationally challenging, with the problem classified as PSPACE-complete, meaning it requires exponential time in the worst case even on polynomial-sized boards.^[34] Hex has fostered a vibrant community, serving as the basis for international tournaments such as those at the Computer Olympiad, where AI programs compete on standard boards.^[35] Notable software includes MoHex, a Monte Carlo tree search-based AI that has dominated these events since 2009 by leveraging pattern recognition and virtual connections for efficient evaluation.^[36] The game has also inspired variants like Misère Hex (or Rex), where the player who connects their sides loses, altering strategic priorities while retaining the core topology.^[37] Its linear connection mechanic on a rhombic board influenced subsequent games, such as Y, which generalizes side connections to three pairs without predefined edges.^[13]

Havannah

Havannah is a two-player abstract strategy connection game invented by Dutch designer Christian Freeling in 1979.^[38] It was first published in 1980 by Ravensburger in Germany, featuring a standard board suitable for beginners.^[18] The game draws inspiration from hexagonal connection games like Hex but introduces diverse winning conditions to enhance strategic depth.^[39] After Ravensburger ceased production, Havannah boards became available through custom makers such as Hexboards, with print-on-demand options emerging post-2022 for larger variants.^[40] The game is played on a hexagonal board composed of smaller hexagons, typically with 8 cells along each side, resulting in 217 playable positions.^[18] Players alternate turns placing a single stone of their color—black or white—on empty cells, with no movement, capture, or removal of pieces allowed.^[41] The objective is to form one of three distinct connected chains with one's stones: a bridge, linking any two of the board's six corners; a fork, connecting any three of the six sides (noting that corners do not count as part of the sides); or a ring, a closed loop of at least six stones encircling one or more empty cells or opponent stones.^[18]^[38] These chains must be continuous via orthogonally adjacent stones (sharing a full edge). The first player to achieve any of these formations wins immediately, and the game is balanced for impartial play with no first-player advantage in optimal conditions.^[42] Havannah's multiple winning paths—bridge, fork, or ring—promote varied strategies and high replayability, as players must defend against threats in multiple directions simultaneously, unlike single-objective connection games.^[39] This complexity has led to its inclusion in abstract strategy tournaments since the late 1990s, including online events on platforms like Abstract Play.^[43] The game's strategic depth has also attracted AI research, with implementations using Monte Carlo Tree Search (MCTS) achieving strong performance; for instance, enhanced MCTS variants with playout pruning have demonstrated competitive play against human experts.^[44] Today, Havannah remains accessible digitally via sites like igGameCenter and BoardSpace.net, alongside physical sets from specialty producers.^[41]^[45]

Tak

Tak is a two-player abstract strategy game in the connection genre, played on a square board with the standard size being 5x5. The objective is to form a "road"—an unbroken path of one's own flat stones or capstones connecting any two opposite edges of the board, akin to the connection goal in Hex. Players begin by placing one of their opponent's flat stones on their first turn, then alternate placing their own pieces: flat stones (which can stack), standing stones (which block stacking and do not count toward roads), or capstones (which count toward roads and can flatten standing stones). Stacks form when flat stones are placed atop existing pieces, limited to a height of four, and only stacks topped by one's own piece can be moved in straight lines, dropping at least one piece per space traversed up to the board's width.^[26]^[46] The game's mechanics emphasize control through stacking and captures. Captures occur when a player moves a stack onto an opponent's stack, gaining control if their piece ends on top; capstones enable flattening of isolated standing stones or placement atop stacks for strategic dominance. Roads cannot include standing stones or diagonal connections, promoting linear path-building amid blocking and disruption. If a player runs out of pieces or the board fills without a road, the player controlling more stacks with flat stones on top wins a flat victory; simultaneous road formation awards the win to the active player. These rules create high tactical depth, as evidenced by AI solvers like Tactician, which explore extensive move trees to evaluate positions.^[26]^[22] Tak originated as a fictional game in Patrick Rothfuss's 2011 novel The Wise Man's Fear, part of the Kingkiller Chronicle series, where it is depicted as an elegant strategy diversion. Rothfuss collaborated with game designer James Ernest to develop a playable version, with mechanics finalized in 2014–2015 and the board game released in 2016 by Cheapass Games following a successful Kickstarter campaign that raised over $1.3 million from 12,000 backers. The game saw increased popularity after 2020, driven by digital adaptations and community growth, including mobile apps and the 2021 inclusion as an event in the Mind Sports Olympiad.^[46]^[47]^[48] The Tak community thrives online via platforms like playtak.com, which supports real-time and correspondence play, and through the United States Tak Association (USTA), which organizes regular tournaments. Expansions feature larger boards such as 6x6 or 8x8, increasing complexity with more pieces (up to 50 stones and 2 capstones) while retaining core rules. Additional resources, like the Tak Companion Book by Ernest and Rothfuss, provide historical context and variant rules, further enriching the game's appeal.^[49]^[26]

The Game of Y

The Game of Y is an abstract strategy connection game played on a triangular board tessellated with hexagonal cells, featuring three designated corner points representing the sides to be connected. Players alternate turns placing a stone of their color on an empty cell, aiming to form a continuous path of adjacent stones (sharing an edge) that links all three corner points simultaneously. The game is designed for two players but can accommodate 2-4 players in variants where multiple opponents share the board, with the first to achieve the connection declared the winner; in multi-player setups, colored paths may be used to distinguish individual goals while maintaining the shared three-way objective.^[50]^[51] The game originated in the early 1950s when mathematician John Milnor first described its mechanics as an extension of connection-based abstract games. It was independently invented in 1953 by Craige Schensted and Charles Titus, who developed it during their studies at the Massachusetts Institute of Technology. The Game of Y received wider attention through its inclusion in Martin Gardner's "Mathematical Games" columns and books in Scientific American, where it was presented alongside related combinatorial puzzles to illustrate graph theory concepts.^[52]^[53] A key feature of the Game of Y is its three-way connection mechanic, which demands paths radiating in multiple directions from the center, thereby amplifying blocking opportunities as opponents can interrupt progress toward any of the three targets more effectively than in dual-edge games. This design promotes dynamic interplay on compact boards suitable for rapid sessions, often resolving in under 20 moves. Y evolved briefly from Hex's dual-edge concept by adapting the board to a triangle and unifying the win condition across all edges.^[54]^[13] Variants of the Game of Y include expanded triangular boards of varying sizes (such as side lengths of 13 to 19 hexes) to extend play depth and challenge, as well as implementations with colored paths for clearer multi-player distinction. Digital versions are widely available on online platforms, enabling remote play and automated analysis while preserving the core rules.^[50]

Additional Connection Games

Historical and Early Variants

The earliest known connection game is Lightning, patented on March 29, 1892, by Selchow & Richter, in which players alternately draw and place tiles on an 8x17 grid to form an unbroken line connecting opposite sides while potentially interfering with the opponent's efforts, predating modern examples by several decades.^[55] This format emphasized direct line connections using physical components, laying groundwork for connectivity as a core mechanic in abstract strategy games. In 1932, Parker Brothers introduced Zig-Zag, the first commercial piece-based connection game, played on a board measuring approximately 22 by 22 inches where players place tokens according to dice rolls to form chains between opposite sides while blocking opponents, incorporating chance elements alongside strategic placement.^[56]^[57] The game's rules promoted quick play for two or four players, blending skill and luck. Zig-Zag's use of physical pieces on a structured board marked a shift from tile drawing to formalized equipment, influencing later designs by introducing edge-to-edge linking.^[57] Early variants beyond the pre-1950s era retained these foundational traits while experimenting with path-building. Psyche-Paths, developed in the late 1960s and later reissued as Kaliko in the 1970s and 1980s, involved placing hexagonal tiles with colored path segments to form loops or connections, doubling scores for closed paths and emphasizing edge-matching without a fixed board.^[58] Similarly, Onyx, conceived by Larry Back in 1984 and first detailed publicly in 2000, added capturing mechanics to path connections on a grid, where players could remove opponent pieces to extend their lines, building on simple board setups with innovative disruption rules.^[59] These historical and early connection games were predominantly analog, relying on tiles, basic wooden or cardboard components, or tile sets, which fostered intuitive no-draw outcomes through topological inevitability—one player must eventually connect.^[55] Their influence persists in the genre's emphasis on unbroken paths and blocking, though sparse historical records have led to modern digital recreations via apps to preserve and analyze forgotten rulesets.^[59]

Contemporary and Digital Adaptations

In the early 21st century, several new connection games emerged, building on classic mechanics while introducing novel elements such as edge-claiming and multi-piece interactions. PÜNCT, released in 2001 as part of the GIPF Project, is a prominent example where players claim edges of hexagonal spaces to form connections between opposite board sides, incorporating movement and capture rules that add strategic depth beyond static placement.^[60] Similarly, Connect6, introduced in 2003 by I-Chen Wu, extends Gomoku principles by allowing the first player to place two stones, emphasizing rapid chain formation on a grid to connect opposite edges.^[61] Digital platforms have significantly expanded access to connection games, enabling asynchronous and real-time multiplayer play. Boardspace.net, an online service since the early 2000s, hosts numerous variants including Hex, Twixt, PÜNCT, Lines of Action, and the Game of Y, supporting both human opponents and computer analysis tools for strategy exploration.^[20] Little Golem, another longstanding browser-based server, offers games like Hex, Havannah, Twixt, Connect6, and ConHex, with turn-based matches that accommodate global players and include rating systems for competitive ranking.^[62] These platforms have facilitated the digital revival of older titles like Twixt, originally from 1962, through dedicated sites such as Twixt Live, which provides real-time online matchmaking and mobile-compatible interfaces.^[63] Advancements in artificial intelligence have enhanced digital adaptations, particularly for solving and playing connection games. For Tak, post-2020 developments include sophisticated bots on Playtak.com, such as TakticianBot and Tiltak, which employ minimax algorithms and heuristic evaluations to simulate expert-level play, allowing users to practice against adjustable AI opponents.^[49] Mobile applications have further democratized access; for instance, "Hex: A Connection Game" on Android enables solo play against AI or local multiplayer on hexagonal boards of varying sizes.^[64] While Tak lacks a dedicated official mobile app, its web-based platform supports cross-device play, and community efforts continue toward app development.^[65] Contemporary adaptations increasingly incorporate immersive technologies and community-driven events. Online multiplayer has grown through platforms like Boardspace.net and Little Golem, supporting 3D visualizations in games like PÜNCT, where elevated pieces create layered connections.^[66] Virtual reality integrations, such as the All on Board! platform, allow connection games to be played in 3D environments, enhancing spatial strategy via gesture controls, though specific implementations remain niche.^[67] Tournaments have proliferated in the 2020s via Discord communities, particularly for Tak, with organized online leagues and bot-assisted qualifiers fostering global competition.^[68] Emerging trends include AI-generated variants, where algorithms inspired by combinatorial game theory software analyze board states to propose rule tweaks, and integration with tools like those in algorithmic CGT for optimal play simulation.^[69]

Strategic and Theoretical Aspects

Player Strategies

In connection games, players employ a range of tactical approaches to establish and protect pathways while disrupting the opponent's efforts, emphasizing efficiency in placement and response to maintain initiative. These strategies are broadly applicable across variants like Hex and Havannah, where the core objective is forming an unbroken connection between designated board edges or points.^[70]^[71] Opening moves prioritize center control to maximize flexibility and potential connectivity options, as central placements allow stones to influence multiple directions and avoid early edge commitments that limit adaptability. In Hex, for instance, the center hex represents the strongest initial move, providing symmetry and access to all sides, while edge-adjacent openings like a2 or b2 can provoke a balanced response under the swap rule.^[70]^[72] Players often create initial threats—such as short bridges or potential forks—to force the opponent into reactive blocks, and double threats (two simultaneous connection risks) can compel the defender to address only one, granting the attacker tempo. In Havannah, early "spider" formations in the center build toward safe frames while probing for opponent weaknesses.^[71]^[70] During the midgame, tactics shift to bridging gaps between existing groups and fortifying perimeter defenses to contain the opponent's advances. Bridging involves efficient templates, like diagonal connections in Hex, to link separated stones without wasteful moves, expanding territory while minimizing exposure.^[70] Perimeter defense requires anticipating and blocking key opponent chains, often by placing stones ahead of their "head" to create barriers, as seen in classic Hex blocks that halt bridge extensions. Sacrifice plays, such as placing a stone in a contested bridge midpoint, can lure the opponent into overcommitting resources, allowing the sacrificer to redirect efforts elsewhere or exploit the resulting isolation. In Havannah, establishing an unbreakable "frame"—a partial ring, bridge, or fork—turns defense into offense by forcing the opponent to respond ineffectively.^[73]^[71] Balancing attack and defense is crucial, as overextension—pushing too far without support—leaves groups vulnerable to cuts and reversals.^[70] Endgame play focuses on path shortening to complete connections swiftly and leveraging virtual connections, where implied links through opponent blocks or bridges act as if already formed for strategic planning. In Hex, identifying the opponent's shortest remaining path and targeting its weakest link ensures a timely win, while virtual bridges treat distant groups as connected if an unblockable path exists between them.^[70] Havannah endgames often devolve into races, where players count moves to finish a frame, prioritizing speed over perfection in balanced positions.^[71] Genre-wide tips include utilizing the pie rule to counter first-player advantage, allowing the second player to swap positions after the opening move for fairness, as commonly implemented in Hex and Havannah.^[30] Maintaining equilibrium between aggression and caution prevents common errors like excessive blocking, which cedes initiative, or ignoring global board threats in favor of local skirmishes. In Tak, for example, early influence-building mirrors these principles by layering flats to control central space without premature road commitments.^[74]^[71]

Mathematical Analysis

Connection games, particularly pure connection variants like Hex, exhibit a fundamental topological property: on a finite board, exactly one player must win, with draws impossible. This no-draw theorem arises from the board's structure, modeled as a planar graph where player connections correspond to paths between opposite boundaries; a full occupation forces at least one path to exist due to the Jordan curve theorem's discrete analog, preventing simultaneous blocks without violating planarity.^[75] David Gale demonstrated that this property is logically equivalent to Brouwer's fixed-point theorem in two dimensions, by mapping Hex outcomes to continuous functions on the unit square and showing mutual implications through fixed-point existence and path intersection guarantees.^[33] The computational complexity of determining the winner in connection games is high; for Hex, the problem of evaluating a given position on an n \times n board is PSPACE-complete, as it generalizes to path-finding in graphs where optimal play requires exploring exponential state spaces.^[34] This hardness extends to many connection variants, including Havannah and TwixT, where deciding connectivity under adversarial play reduces from known PSPACE-complete problems like generalized geography.^[76] Due to this complexity, exact solutions are infeasible for large boards, leading AI approaches to approximate strong play via Monte Carlo tree search (MCTS), which builds game trees through random simulations and has powered top Hex programs like MoHex.^[77] Key theoretical concepts in connection games include equivalence to the Shannon switching game, where Hex on a hexagonal grid corresponds to a vertex-coloring variant on its dual graph, allowing Short (the connector) to claim vertices and Cut to block, with winning conditions mirroring edge contractions and deletions.^[78] Graph duality further aids analysis: the opponent's connection paths in the primal graph become cuts in the dual, enabling proofs of strategy symmetry and non-planar contradictions in supposed draw positions.^[75] Research on connection games has advanced through solving small boards and exploring equilibria in variants. Computational methods, including decomposition into local patterns and virtual connections, have solved 7×7 Hex, revealing first-player winning openings and 14 second-player losing responses under perfect play.^[79] In imperfect-information variants like Dark Hex, where players observe only their moves, finding approximate Nash equilibria via counterfactual regret minimization highlights strategic depth, with equilibria yielding balanced utilities despite hidden information.^[80]

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Mathematical Games - Scientific American
July 1, 1957. 1 min read. Mathematical Games. Concerning the game of Hex, which may be played on the tiles of the bathroom floor. By Martin Gardner. Join Our ...
[17]
Bridg-It - HexWiki
Aug 4, 2024 · It was invented by the mathematician David Gale, and was described, under the name "Gale", in Martin Gardner's October 1958 column in Scientific ...
[18]
Y - HexWiki
Aug 27, 2025 · The game of Y is a connection game first invented in the early 1950s by John Milnor, and independently discovered in 1953 by Craige Schensted ( ...
[19]
Twixt | Board Game - BoardGameGeek
In stock Rating 3.3 (2,642) Twixt is a 2-player abstract game invented by Alex Randolph. TwixT is a connection-type game, very similar to Hex.
[20]
History of TRAX - Traxgame.com
TRAX was invented by a New Zealander, David Smith, and first published in that country and the United States in 1982. Favourable reviews, prestigious awards and ...
[21]
Dale Yu: First Impressions of Tak | The Opinionated Gamers
Apr 24, 2016 · Tak is a fictional game found in a book that has since come to life. The game was first described in passing in The Wise Man's Fear, a 2011 novel by Patrick ...
[22]
Games - Cameron Browne
I've always had a keen interest in abstract board games, puzzles and mathematical recreations in general, and started designing my own board games around 2000.
[23]
Havannah | Board Game - BoardGameGeek
Rating 7.2/10 (215) Havannah is a connection game (Hex, Twixt, etc.) with several interesting twists. The board is a hexagonal grid shaped in the form of a hexagon--8 hexes per ...
[24]
Combinatorial game theory - HexWiki
May 12, 2024 · Combinatorial game theory is a formalism that can be used to analyze sequential two-player perfect information games.
[25]
BoardSpace.net Home Page
A place to play Board Games. All these games can be played with friends - online or offline - desktop or mobile. How to Play Online Games here ...Hive · Fanorona · Trike · Circle of Life
[26]
Analysis of the length of optimal games of Hex game using ...
Sep 8, 2024 · The following results come from a strong Hex AI called KataHex. The AI and the raw data with visualization scripts are here: KataHex release.<|control11|><|separator|>
[27]
https://www.sciencedirect.com/science/article/pii/S030439751630278X
[28]
[PDF] On the Complexity of Connection Games
Mar 11, 2016 · In these games, players take turns placing or/and moving pieces until they connect the two sides of the board. Hex, y, and twixt are typical ...
[29]
[PDF] EfficientPlayouts for the Havannah Abstract Board Game
The board game Havannah was invented by Christian Freeling in 1979 and pub-.Missing: publisher Winther
[30]
[PDF] Object of the Game The Board - racingcow.io
You can play Tak on many different sizes of board, from 3x3 to. 6x6 and even larger. This game includes a double-sided Tak Board: an elegant Selas. Flower board ...
[31]
On the complexity of connection games - ScienceDirect.com
Sep 6, 2016 · Connection Games: Variations on a Theme. AK Peters (2005). Google Scholar. [3]. B. Arneson, R.B. Hayward, P. Henderson. Monte Carlo tree search ...Missing: early origins
[32]
[PDF] HEX 1. Introduction The game of Hex was first invented in 1942 by ...
The Hex Theorem states that a game of Hex cannot end in a draw. The only way ... David Gale gave a simple proof of the. Hex Theorem based on graph theory and also ...
[33]
Hex Can't End in a Draw
David Gale has even shown that the fact that Hex can't end in a draw is equivalent to Brouwer's Fixed Point theorem. There is a proof by induction (D.
[34]
Hex - igGameCenter
This game is a member of the connection game family , including such games as Y , Havannah and Gonnect . ... pie rule is generally used to make the game fair.
[35]
Rules
It is a two-person zero-sum game with perfect information. LOA is a connection game, albeit non-typical. Claude Soucie invented it around 1960.
[36]
Game of Hex -- from Wolfram MathWorld
Hex is a two-player game invented by Piet Hein in 1942 while a student at Niels Bohr's Institute for Theoretical Physics.
[37]
Rules - HexWiki
Jan 20, 2023 · To balance the game, the "swap rule" or "pie rule" is used at the start of the game. The first player places a red stone on the board, and then ...
[38]
[PDF] the game of hex and the brouwer fixed-point theorem
Theorem is evident. We wish to emphasize, however, that the analysis to follow and the relation to the Brouwer Theorem depends only on the no-draw property.
[39]
[PDF] Hex is PSPACE-complete -. [ Tim Wylie ] .
In this paper we will show that the same holds for the game of Hex. The crucial point of the proof is to establish PSPACE-hardness for a gen- eralization of ...
[40]
https://www.etsy.com/shop/Hexboard
[41]
MOHEX Wins Hex Tournament - Semantic Scholar
MoHex, the Monte Carlo tree search Hex player that won gold at the 2009 Computer Olympiad is described and its main contributions to Monte CarloTree search ...
[42]
[PDF] Who Wins Misère Hex?
The purpose of this note is to analyze a variant of Hex that we call. Misère Hex. The difference between normal Hex and Misère Hex is that the outcome of the ...
[43]
Havannah at Sensei's Library
May 25, 2020 · Havannah is a connection game invented by Christian Freeling in 1979. It is related to the game Hex, but the winning condition is different.Missing: publisher Winther
[44]
Havannah - HexWiki
Jul 12, 2021 · The goal of the game is to be the first player to make a chain that forms one of these winning patterns: a fork: a connected chain that has ...
[45]
Hexboard - Etsy
4.7 1.5K All items · Havannah board sizes 10 and 11 · Game of Y · Modular hexboard 9x9 11x11 13x13 15x15 · Hexboard for the games of Havannah, Atoll, Wings and Unlur.<|control11|><|separator|>
[46]
Havannah - igGameCenter
Overview. Havannah is a two-player abstract game played with black and white stones on a hexagonal board, invented by Christian Freeling in 1976.Missing: publisher Winther
[47]
Havannah - Game rules
Fork: connecting three of the six sides of the board. The corners are not considered part of any of the sides. Ring: connecting the chain with itself, so that ...
[48]
Full Report for Havannah by Christian Freeling - Stephen Tavener
I'm surprised more fans of abstract games don't play this. I made my own wooden board. Fascinating game! morphles, 10, Though more complicated rules than hex.
[49]
[PDF] Pruning playouts in Monte-Carlo Tree Search for the game of ... - HAL
Jul 5, 2016 · We experiment our improvement, called “Playout Pruning with Rave” (PPR) on the game of Havannah. Classic MCTS algorithms already provide good ...
[50]
About Havannah - BoardSpace
There are 3 ways to win; forming a "bridge" connecting two corners, forming a "fork" connecting 3 sides, or forming a "ring" which is any closed loop. Draws are ...
[51]
Tak Board Game - Crab Fragment Labs
Tak is a two-player game that uses a square board and stacking pieces. Players win by creating a road, which is a line of pieces that connects opposite sides ...
[52]
nelhage/taktician: An implementation of and AI for the game of Tak
By default, taktician analyze will search for up to 1m before returning a final assessment. Use -limit 2m to give it more time, or -depth 5 to search to a fixed ...
[53]
Tak: A Beautiful Game by Cheapass Games - Kickstarter
Nov 22, 2021 · Tak is a new abstract strategy game created by James Ernest and Patrick Rothfuss, based on the game in The Wise Man's Fear.Missing: Workshop | Show results with:Workshop
[54]
Where to Play Tak - USTak.org
Where to Play Tak · Online. Playtak.com. Maintained by the USTA. Designed to be an “over the board” experience. Platform where most USTA tournaments are held.
[55]
Play Tak
Play online Tak games. Tak is a strategy board game created by James Ernest and Patrick Rothfuss.Missing: post- 2020
[56]
Y - igGameCenter
"Y" is played on a triangular board with hexagonal cells. The following board sizes are used on igGameCenter: 13x13x13, 14x14x14, 15x15x15, 19x19x19. Play.Missing: configuration | Show results with:configuration
[57]
[PDF] arXiv:2102.02621v1 [math.CO] 4 Feb 2021
Feb 4, 2021 · Game of Y: The board has 3 boundary paths l1,l2,l3. The goal for ... [Gar59] Martin Gardner, The Scientific American Book of Mathematical Puzzles ...
[58]
(PDF) Hex implies Y - ResearchGate
Feb 4, 2021 · We give a simple and short proof of the fact that the board game of Y cannot end in a draw. ... Martin Gardner's ﬁrst book. of mathematical ...
[59]
Martin Gardner Fans: Try These Mathematical Games
Oct 1, 2014 · Try these mathematical games. Enjoy five puzzles posed in "Let the Games Continue," a celebration of Martin Gardner's legacy.Missing: Y | Show results with:Y
[60]
Hex and combinatorics - ScienceDirect.com
Oct 6, 2006 · Like Hex, Y was discovered on more than one occasion, including by John Milnor ... The Game of Y: a game won by Black (left) and Hex as a special ...<|control11|><|separator|>
[61]
[PDF] Hex variants • A beautiful move in Renju • Anchor
Lightning A Connection Game from the 1890's by Jim Polczynski. 10 A Beautiful Move in Renju by Ants Soosyrv. 12 Lines of Action Strategic Ideas -- Part 4 by ...<|control11|><|separator|>
[62]
[PDF] 1932 • The Parker Games and Amusements
223X x22^ inches, and complete directions for play. Price $1.25. ZIG ZAG. Reg. U. S. Patent Office. (New).
[63]
https://twixtlive.com/
[64]
Kadon Enterprises, Inc., Games Magazine reviews of Kaliko
This boardless path-connecting game is an improved version of Psyche-Paths, a little-known but highly original game of the 1960s.
[65]
[PDF] First of a new series on Hostage Chess • Bao: the King of Mancalas
• Onyx: a brilliant new connection game. • First of a new series on Hostage ... I realized back in 1984 that there was a dearth of connection games with ...
[66]
PÜNCT | Board Game - BoardGameGeek
In stock Rating 3.4 (1,920) PÜNCT, the fifth released game by design in project GIPF, is a connection game. The goal is to connect two opposite sides of the board.
[67]
Connect6 - LG-Docs
Jul 30, 2021 · Connect6 introduced in 2003 by Professor I-Chen Wu. Rules: The rules of Connect6 are very simple and similar to the traditional game of Gomoku.
[68]
Little Golem
No information is available for this page. · Learn whyMissing: connection online platforms Boardspace.
[69]
TwixT Live - Play
You need to use one of the following browsers to play TwixT on this site: Mozilla Firefox, Google Chrome, Safari, Opera, or Internet Explorer 9 or higher.
[70]
https://www.hexwiki.net/index.php/Basic_%28strategy_guide%29
[71]
'Tak' Mobile Version For iOS/Android - Reddit
Dec 12, 2023 · I would just do it to help build my portfolio and because I love the game so much. I might just throw it up on the App Store with a different ...
[72]
Punct at Boardspace.net
Pünct is a connection game which includes some 3D elements. The primary goal is to form a chain connecting two opposite edges of the board, but the rules ...
[73]
All On Board! – Enjoy Board Games in XR! - The Game Kitchen
All on Board! is a revolutionary tabletop platform built from the ground up specifically for VR. As the only tabletop platform natively built for VR.
[74]
The Discord Community & Online Play: Not your standard Tak review
Nov 28, 2022 · Playing Tak online is free, whether you sign on as a guest with one click or create your own account on the PlayTak website.Missing: solvers | Show results with:solvers
[75]
[PDF] Algorithmic Combinatorial Game Theory - Erik Demaine
Algorithmic Combinatorial Game Theory analyzes ideal play in perfect-information games, often with two players, and explores the complexity of determining ...Missing: software | Show results with:software
[76]
Basic (strategy guide) - HexWiki
Oct 12, 2022 · Playing in the center hex is the strongest opening move. The weakest opening move is to play in one of the acute corners (a1 and the opposite ...Rules of Hex · Basic strategy · General principles · The opening
[77]
Strategy - MindSports
The main strategic goal in Havannah is is the establishment of a frame, a connection aiming at a ring, bridge or fork, that, though still incomplete, cannot be ...Missing: publisher | Show results with:publisher<|control11|><|separator|>
[78]
Connection Games I: Hex | Dr Eric Silverman
a simple goal, easy to see and easy to ...Missing: definition | Show results with:definition
[79]
Connection Games III: Havannah and Starweb | Dr Eric Silverman
Mar 3, 2020 · Havannah is a connection game that offers a completely unique take on the genre. Connection games are typically characterised by a sense of absolute clarity.Missing: Winther | Show results with:Winther
[80]
Swap rule - HexWiki
Jul 21, 2025 · The swap rule is sometimes also called the "pie rule", since it resembles the well-known "you cut, I choose" method for fairly dividing a pie ...
[81]
[PDF] the game of hex: a study in graph theory and algebraic topology
We have in fact stated the Hex theorem in a slightly stronger form than it is usually stated; in most contexts the Hex theorem only guarantees that G is a win.
[82]
[PDF] On the Complexity of Connection Games
proof that Shannon's vertex switching game is PSPACE-complete [14] and. Reisch's proof that hex is PSPACE-complete [15], the only complexity results on ...
[83]
[PDF] Monte Carlo Tree Search in Hex - Department of Computing Science
In this paper we describe MoHex, the Monte Carlo Tree Search. Hex player that won gold in the 2009 Computer Olympiad. Our main contributions to Monte Carlo ...
[84]
[PDF] playing and solving the game of hex
The game of Hex is of interest to the mathematics, algorithms, and artificial intelligence communi- ties. It is a classical PSPACE-complete problem, and its ...<|control11|><|separator|>
[85]
[PDF] New Winning and Losing Positions for 7×7 Hex
In this paper we apply a decomposition method to obtain a new winning strategy for 7x7 Hex. We also show that some positions on the 7x7 Hex board, called ...
[86]
http://webdocs.cs.ualberta.ca/~hayward/theses/bedir.pdf