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Using Graph Theory to Outsmart Opponents Ken DelSanto May 6 th 2011 Graph Theory Conference Dr. A. Beecher Ramapo College of New Jersey

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Overview 1.Chess A. Knight B. Bishop/Rook C. King/Queen 2. Risk A. Map/Board B. Territories C. Cut-Sets

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Chess Definitions File : On a chessboard, the files are the columns, lettered a-h from left to right. Rank : On a chessboard, the ranks are the rows, numbered 1-8 from bottom to top. Tour : A path around the chessboard where the piece touches every single square exactly one time. Closed Tour : A path around the chessboard where the piece touches every single square exactly once, beginning and ending at the same square. Promotion : The action in which the pawn reaches the 8 th rank, and is allotted to choose to become another piece (usually a queen). Royal Pieces/Attacking Pieces : Royal pieces are the king and queen(s). Attacking pieces are bishops, knights, and rooks.

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Chess The Knight - The knight’s graph (right) shows the number of possible moves for a knight from any space on the board. - Knowing this graph can help you plot out or visualize your attacking strategy, or what your opponent intends to do. - If the knight is one of your last (or your opponent’s) piece remaining, knowing this graph can help you promote your pawns, or attempt to prevent promotions.

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Chess The Knight’s Tours - Knight’s tours (bottom) are not trivial. King, queen, rook tours are easy to see. - Knight’s closed tour (top) discovered in 1770 by machine, some 700 years after the game’s origin. - The knight’s tour closely relates problems in graph theory regarding Hamiltonian Paths.

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Chess The Bishop and the Rook - Rook’s graph (right) is a regular graph where each node has degree 14. - The rook’s (king’s and queen’s) tour is easy to see from any point on the chess board. - The bishop cannot have a tour, being it only can move to half the board.

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D(Rook) = (14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14,14, 14, 14, 14, 14,) D(Bshp) = (13, 13, 12, 12, 12, 12, 11, 11, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6) D(Kght) = (8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2) Chess The Great Debate! - For years chess players have debated which attacking piece is most valuable, and which is least. - Using degree sequences (right), we can provide an interesting argument. - Perhaps…#1 Rook, #2 Knight, #3 Bishop? - Perhaps…#1 Rook, #2 Bishop, #3 Knight?

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Chess The King and the Queen -The king’s graph (right) is actually a subgraph of the queen’s graph. - Additionally, extend each edge to every node in every direction (horizontally, vertically, and diagonally) and we can obtain queen’s graph - By looking at all the graphs, we can see that it is impossible to checkmate an opponent with solely a bishop and king, or solely a knight and king. It is possible to checkmate an opponent with just a rook and a king (and therefore obviously a queen and a king).

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Risk Definitions Territory : On a Risk game board, a territory is a section of the map that is pre-designated by the game creator. Cut-Set : Set of vertices or edges in a graph that, if removed from the graph, the graph is broken into more components than the original graph. Cut-Point : A cut-set consisting of a single vertex. Bridge : A cut-set consisting of a single edge.

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Risk The Game Board - The map (right) is easier to visualize than the game board. - Using the territories as vertices, and the borders between territories and edges, we can create a graphic version of the game board of Risk. - It is easier to analyze the graph if the continents are color coordinated. That way, we can understand cut-sets and degrees of vertices easier.

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Risk The Graphic Game Board - When analyzing this graph, we can see the best way to conquer the world by finding the higher degree territories. - Additionally, it is easy to spot what edges (or borders) connect the continents. This will help us in finding our cut-sets. - Finally, finding subgraphs within continents can help us figure out how to conquer certain continents.

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Risk Territories D(GameBoard) = (6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2) - The degree sequence of the game board (right) shows us the degree of each territory on the game board. - Clearly, the territories that have higher degrees are more important when trying to take over the world! - In addition, the territories that we will consider part of our cut-sets are important to taking over certain continents.

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Risk Cut-Sets, Cut-Points, and Bridges - To conquer a continent, all we need to do is simply block the entrances from other continents. - We can see that these a the cut-sets of the graph of the game board. - Now we can see that some continents are disconnected from others. These territories are essential to conquering the continent.

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Now GO CONQUER THE WORLD! or CAPTURE THE KING!

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- McFarland, Thomas. The Logics of Chess. University of Wisconsin Whitewater. 2010. http://math.uww.edu/~mcfarlat/177.htm - Smith, Ethal. The History of the Board Game Risk. Helium. 2007. http://www.helium.com/items/376520-the-history-of-the-board-game-risk - Nonenmacher, R.A. Knight's Graph Showing the Number of Possible Moves from Each Node. June 2008. - Dharwadker, Ashay. Pirzada, Shariefuddin. Applications of Graph Theory. Journal of the Korean Society for Industrial and Applied Mathematics. 2007. http://www.dharwadker.org/pirzada/applica - Smith. 8x8 Rook's Graph. April 2008. - Elkies, Noam. Chess and Mathematics. Harvard University. 2004. - Smith. 8x8 King's Graph. April 2008. - Crowley, Patrick. Risk Territories. Graffletopia. April 2009. - Pinto Michael. Risk: Conquering Inner Napoleans Since 1959. Fanboy. April 2010. - Mittal, Aditya. Chai, Yi. Wang, Tiyu. Yared, Elie. Wang, Qian. Modeling the Attack Phase in Risk: The Board Game Using Graph Theory. Standford University. 2004. Sources

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