Some Mathematics in the Game Polygon

Background on PolygonBoard games inspire friendly competition and promote the ability to strategize on the spot.Additionally, the study of board games is intertwined with the study of mathematics, and numerous research projects have endeavored to explore that connection.Some examples include the applications of graph theory, combinatorics, and computer science to games such as Blokus, Catan, Nim, and]. Yet again, there are contexts like the game of Sylver coinage, the game of chomp, and the postage stamp problem that mirror the formation and properties of mathematical structures such as posets, monoids, and numerical semigroupsPolygon is a strategy board game previously manufactured by Jax, Ltd. starting in 1998.Since then, the game has fallen out of production, and the only copies that are available for purchase appear to be from e-commerce vendors.Despite its obscurity, Polygon is an interesting game to examine through the lens of mathematics.Combinatorics, probability, and algebraic structures play key roles in gameplay and strategy. GameplayPolygon is played on a hexagonal board subdivided into smaller hexagons with 2-4 players placing 49 hexagonal tiles on the board, with the ultimate objective of obtaining the highest total score at the game's end.The values and counts for each type of tile are listed in Table 1.The game begins with players mixing up and laying out all 49 hexagonal tiles face down so that their values are hidden.Each player draws a tile and places it face up on the board.The player with the highest-value tile starts the game, where no distinction is made between regular tiles and multiplier tiles (i.e., those containing a "×"), and Wild tiles (which we label here as "W") are considered to have the highest value.The player with the highest-value tile places it face-up on the space marked START on the game board, located directly in the center of the board.A black-and-white version of the game board is shown in Figure 1.For readability, we have not labeled the START tile as such, and we display all board spaces and tiles so that the text is right-side-up.