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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 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 Smith, Ethal. The History of the Board Game Risk. Helium Nonenmacher, R.A. Knight's Graph Showing the Number of Possible Moves from Each Node. June Dharwadker, Ashay. Pirzada, Shariefuddin. Applications of Graph Theory. Journal of the Korean Society for Industrial and Applied Mathematics Smith. 8x8 Rook's Graph. April Elkies, Noam. Chess and Mathematics. Harvard University Smith. 8x8 King's Graph. April Crowley, Patrick. Risk Territories. Graffletopia. April Pinto Michael. Risk: Conquering Inner Napoleans Since Fanboy. April Mittal, Aditya. Chai, Yi. Wang, Tiyu. Yared, Elie. Wang, Qian. Modeling the Attack Phase in Risk: The Board Game Using Graph Theory. Standford University Sources

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