Black Path Game

Lore & Background

The Black Path Game was invented in 1960 by Larry Black, who may also be known as William L. Black. At the time, Black was an undergraduate at the Massachusetts Institute of Technology, investigating the games Hex and Bridg-It, which are based on creating a connected chain of counters linking opposite sides of a board. His research led to a new topological game that his friends called Black. The game was later introduced to a wider audience by Martin Gardner in his October 1963 'Mathematical Games' column in Scientific American.

The game is played on a board ruled into squares, with one edge designated as the start of the path. Players alternately extend the path by filling the adjacent square at the end of the current path with one of three tile configurations, each containing two paths linking two sides. The first two tiles are those of the Truchet tiling. The path may return to a previously filled square and follow its unused segment. The player who first causes the path to run back into the edge of the board loses.

Strategy for the game was developed by Black's friend Elwyn R. Berlekamp, who discovered a domino tiling strategy. On a rectangular board with an even number of squares (at least one side-length even), the first player has a winning strategy by always playing so the end of the path falls on the middle of a domino. On a board with an odd total number of squares (both sides odd), the second player can win using a similar domino tiling strategy that occupies every square except the one containing the first player's first move.

Reader's Guide

The Black Path Game is significant as an early example of a topological connection game, emerging from the same MIT environment that produced Hex and Bridg-It. Its analysis in Winning Ways for your Mathematical Plays and its presentation by Martin Gardner helped cement its place in recreational mathematics. The game's strategy, discovered by Elwyn R. Berlekamp, uses a domino tiling approach that provides a clear winning condition for either player depending on board dimensions. This strategy illustrates how mathematical tiling concepts can be applied to game theory. The game also shares design elements with later tile-based games such as Tantrix, Trax, and Tsuro, which use similar square or hexagonal tiles with multiple paths. The Black Path Game remains a classic example of a simple but deep two-player game, often studied for its elegant combinatorial structure and its connection to the Truchet tiling.

Did You Know?

• The Black Path Game was invented in 1960 by Larry Black, an undergraduate at the Massachusetts Institute of Technology.
• The game was introduced to the public by Martin Gardner in his October 1963 'Mathematical Games' column in Scientific American.
• The first two tile configurations used in the game are the tiles of the Truchet tiling.
• Elwyn R. Berlekamp, a friend of Black, discovered the domino tiling strategy for winning the game.

Origins in a MIT Classroom

The Black Path Game traces its creation to 1960, when a young undergraduate at the Massachusetts Institute of Technology named Larry Black (also reported in some accounts as William L. Black) was deep in research on two existing topological games: Hex and Bridg-It. Both of those games center on the challenge of building a connected chain of counters that links two opposite edges of a playing board. Black's creative work in that space produced an entirely new topological game, one that his friends — perhaps with a touch of unimaginative brevity — simply called "Black." The game went by several other names over the years, including "Brick," and was eventually described and formally analyzed in the well-known mathematical text Winning Ways for your Mathematical Plays. What began as a student's exploratory exercise in connectivity and topology would go on to capture the attention of the broader mathematical and recreational-gaming community.

The Mechanics of a Growing Path

The Black Path Game unfolds on a grid of squares, with one boundary edge pre-designated as the origin of the path. The opening move establishes the starting point, and from there the two players take turns extending the path outward. Each move consists of placing a tile in the square immediately adjacent to the current end of the path. The available tiles come in exactly three configurations, each containing two internal paths that connect two sides of the square. The first two of these tiles are recognizable as the classic Truchet tiling pieces. A distinctive rule allows the path to loop back into a square that has already been filled, picking up the segment of path that was not yet used. The game ends the moment a player's move causes the path to touch the outer edge of the board — that player loses. This elegant combination of simple placement and a single losing condition creates a deceptively deep tactical landscape.

Dominoes, Diagonals, and the Mathematics of Winning

The strategic heart of the Black Path Game lies in a pairing strategy built around domino tiling. On any rectangular board where at least one dimension is even — meaning the total number of squares is even — the first player holds a guaranteed winning method. The idea, discovered by Black's friend Elwyn R. Berlekamp and later published in his book, is to mentally cover the board with 2×1 dominoes and always place a tile so the path's endpoint lands on the middle of a domino rather than its edge. This forces the opponent to eventually push the path into the board's boundary. When both dimensions are odd, the roles reverse: the second player adopts an analogous domino strategy, leaving only the square of the first player's opening move uncovered. A further subtlety involves the "split domino" — a pair of squares created when a tile is placed on the main diagonal — which the second player can exploit to maintain control. Routing the path into a corner is also a decisive winning tactic, since the opponent is then forced to make the losing edge move.

Gardner's Column and a Family of Path Games

The Black Path Game leapt from the small circle of MIT students into the wider public consciousness in October 1963, when Martin Gardner featured it in his celebrated "Mathematical Games" column in Scientific American. That single article introduced the game to a vast readership of puzzle enthusiasts, teachers, and mathematicians, and it remains the primary public reference point for the game's rules and example positions. The game also occupies a natural place within a broader family of path-building board games. Tantrix uses hexagonal serpentiles with three colored paths and six entry points per tile. Trax employs square tiles with two colored paths and four entry points, closely mirroring the Black Path Game's tile set. Tsuro expands the concept further with four paths and eight entry points on each square. Together, these games form a spectrum of topological path puzzles, and the Black Path Game stands as the foundational two-player square-tile entry in that lineage.

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