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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. Overview. Chess. 2. Risk. Map/Board Territories Cut-Sets. Knight Bishop/Rook King/Queen. Chess. Definitions.
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Using Graph Theory to Outsmart Opponents Ken DelSanto May 6th 2011 Graph Theory Conference Dr. A. Beecher Ramapo College of New Jersey
Overview • Chess 2. Risk • Map/Board • Territories • Cut-Sets • Knight • Bishop/Rook • King/Queen
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 8th 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.
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.
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.
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.
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? 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 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).
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.
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.
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.
Risk Territories • 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. 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)
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.
Now GO CONQUER THE WORLD! or CAPTURE THE KING!
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