Abstract (m,n,k)-Games and Combinatorial Game Theory Interesting Facts about the Combinatorics and Game Theory: In this lesson plan we introduce the generalized version of everyone’s first and favorite pencil and paper game tic tac If we ignore symmetries of the board there are: 131, 184 finished toe, (m, n, k)-games. (m, n, k)-games are games in James McErlain games are won by X; 77, 904 finished games are won by O; 46, 080 which are played on a (m, n) grid with the winning 2 finished games are drawn. And with symmetries: 91 unique positions condition being first to k in a row. (CM) NSF Graduate Fellow are won by X; 44 unique positions are won by O; 3 unique positions We motivate the game by having the students play tic tac are drawn. toe and then expand the game into various other dimensions San Francisco State Compare this with the upper limit of possible games in Go, which is and finally analyzing the game using various base cases and 171 University 1010 . This is far more than the number of atoms in the universe, initial game states and end game states and finally we talk which is around 1082. This is why it is practically impossible to about the game trees for the various games and the total program a computer to play Go well (experienced little kids can beat possible number of games that can be played. A little history: the most advanced computers). This is not the case for Chess, however This lesson was part of a large unit on combinatorial game where super computes can beat and tie and lose to even the best The games we will be talking about though today are what are known as combinatorial games. theory in students were introduced to games such as nim players (i.e., they are fairly on par). and two pile nim, as well as various versions of sprouts, etc. In these games there is no change, or randomness (i.e. dice, etc.) and we always see what our This lesson is intended for middle school and high school opponent is doing. There is also no diplomacy, you are always trying to beat your opponent. The Results, The Experience and Extensions: students. Chess and checkers are games of this nature as well as Go and Mancala and Blockus. These games are studied by mathematicians because we can program computers to play them and thus The classes and groups I have implemented the lesson with seem to Learning Objective develop strategies for playing them using mathematics. We do this by creating what are called really enjoy it. It is clearly accessible to everyone who has played tic tac Students will be able to play various (m, n, k)-games and game trees. toe and even if they haven’t the rules are simple enough while the create a game tree to analyze base cases and end game strategies and solvability of the various game states are simple enough states. They will then use game trees to form strategies. for elementary school students to master. Possible Extensions: (m, n, k, j)-Games, i.e. 3 dimensional games or Definitions tic tac toe on a torus. Definition 1: (m,n,k)-game: games played on a (m, n) board with k in a row being the winning gaming state. Definition 2: Game tree: a diagram that shows the various states and options of a game. Definition 3: Strategie: a technique for winning a game. Definition 4: Solvable Game: a game in which we can create strategy/algorithm that will guarantee a win or a draw for one of the players. Definition 5: Algorithm: a sequence of steps that we can tell a computer to do. Figure: Gomoku: an example of a (19, 19, 5)-game Definition 6: Gomuku: an (m, n, k)-game with Acknowledgments and Funding (19, 19, 5). This game is played on a Go board and his Procedure Dr. Gubeladze, advisor very popular in Japan. It is even played professionally with Begin by having the students play a few rounds of Tic Tac Toe with various people all while Dr. Beck and Dr. Weigners, (CM)2 advisors a variation to the opening sequence to account for first keeping track of who is going first and who is winning. After awhile the students will see that person advantage. Ann Lyon and Robert Pablo; My partner teachers at Thurgood either the first person will win or in most cases there will be a draw. Next pass out the tree Marshall High School Game Tree diagram for the a game of tic tac toe and discuss the symmetry of the tree and how it can be My wife and son for putting up with me further reduced. Supported by San Francisco State UniversityOs˜ (CM)2 program, a We will now introduce the more generalized (m, n, k)-game. Ask the students to try playing a branch of the NSF GK-12 program (Grant DGE-0841164) game of tic tac toe with only getting 2 in a row, or playing on a 4x4 board and getting 3 in a row. Have them experiment with small versions of this game. Are there differences in how you play (Strategies)? Does someone always win/lose? Next we will look at base cases of games. Have the students play a game of (m, 1, 2) or (m, 1, 3). What do they notice? Can they create a game tree for these games? What about a (m, 2, 2) or (m, 2, 3). Continue this this manner until they have reached a full board (n, n, k). What do they notice as they build up the board and expand in one direction? When do they know there will be a winner? Is it before the last piece is played? Game Tree Next have the students play a game of gomuku! (19, 19, 5) and see what if any strategies the learned previously they can apply to the much more difficult version. Is there still an advantage Figure: Graphic taken from: http://www.cynical-c.com/ for either player going first? When do you know the game is won? etc.