Vellore Institute of Technology

Graph Theory: MAT206Slot: C2+TC2

Applications of Graph Theory inBoard Games

By:12BCE0033Abhinav Prasad

Supervisor:Prof. Madhu Sudhan

Reddy

April 22, 2015

1 Introduction

Board games have existed since the 1800s as a means of entertainment for themasses. The Board games of old featured mind boggling challenges and puzzleswhich would take hours for any man to solve. Modern Applications of GraphTheory to board games help solve these games efficiently and provide optimalstrategies to win the game. In this text, we will look at one of the old boardgames: Hex.

2 Hex

Hex is a simple to learn but difficult to solve board game invented by Piet Hien,A Danish mathematician in 1942. It can be considered as a strategy board gameplayed on a hexagonal grid traditionally on a 11x11 rhombus.

2.1 Rules

Hex is played on a rhombus-shaped board consisting of hexagonal shaped spaces.The goal of the two players, black (B) and white (W), are to form a chainfrom one edge of the board to the opposite edge. Bs goal is to make a chainof black pieces connecting B and B, and W wants to make a chain of whitepieces connecting W and W. Below is a picture of the empy game board:

To play the game, the black and white players alternate placing pieces in one ofthe hexagonal cells. There are no restrictions as to where pieces can be played.The first player to create a chain as described above wins. The game ends whena player wins or when the board is filled.

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2.2 Solution

A two player Hex game cannot end in a draw. This can be proved using Ku-ratowskis Theorem. The player who starts first has a nxn winning strategy.However, the second player can covert the first players strategy into their ownby making an irrelevant move and following the first players strategy. Also,placing a cell on the board will either benefit the player or prove useless, butwill never hurt the player.

We can assume the game board as a planar graph G with four corner vertices.Let C be a cell with 6 vertices and 6 edges, and graph G consists of n number ofCells entirely filling it. Thus, each cell C is a region of graph G. Each internalcell is surrounded by 6 other cells, while the corner cells touch 3 cells.

In the above figure, Black has won the game. Thus, black has to make a pathout of Black cells to cross the game board and defeat white. Whites objectivewould be to prevent Black from doing so, by obstructing his way and makinghis own path. An optimal solution to the game can be formulated taking themoves of both players into account and generating a solid winning strategy forthe player.

3 Conclusion

With an understanding of Graph Theory concepts, one can understand theformat of board games and approach these games appropriately. Hex, Sudoku,Snakes and Ladders are examples of Board games which were based on graphtheory concepts which are played even today.

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IntroductionHexRulesSolutionConclusion