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Duo

Duo. By: Fernando & Vivian. GAME DESCRIPTION. a set of ominoes with 21 Blokus tiles:. Two players, alternate turns to place a piece of omino on the board of 14x14 (The starting points will be fixed on the board, somewhere near the center). Valid move:

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Duo

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  1. Duo By: Fernando & Vivian

  2. GAME DESCRIPTION • a set of ominoes with 21 Blokus tiles: • Two players, alternate turns to place a piece of omino on the board of 14x14 (The starting points will be fixed on the board, somewhere near the center) • Valid move: • touches at least one piece of the same color • corner-to-corner contact -- edges cannot touch • Game ends: • no more valid move for both players • Pay off: • Whoever has the least number of squares left wins

  3. GAME CLASSIFICATION • DETERMINATE • NON ZERO-SUM • PERFECT INFORMATION • SEQUENTIAL • NEITHER NORMAL NOR MISERE

  4. RESEARCH QUESTIONS • How many game states does • the Blokus Duo have? • Is this game fair or unfair?

  5. GAME STATES Step 1: Find the size of board corresponding to the ominoes that is offered. Step 2: Start with the smaller size of the board, and find the number of game states. Step 3: Use the combination and the number of corners on the board to estimate the bigger size of the board.

  6. STEP 1: BOARD SIZE Regular game: 14x14; 21 pieces of ominoes; # of squares on the board: 14x14 = 196; # of squares the ominoes have: 89; If the players are offered the free polyominoes of from one to two squares, which is: Then there are 3 squares on the ominoes, Ratio: 3/89 = (x^2)/196

  7. Step 1

  8. Step 2: Game states for smaller board Draw out every game configurations for smaller board. For 3x3 board, 40 game states; For 5x5 board, 2500 game states;

  9. Step 3: combination for Bigger board Formula: C (n, r) = n!/(r!(n-r)!) Assume each piece of omino has four corners If m = the number of moves that has been done for each player c = the number of corners available on the board; c = 4m – (m - 1) = 3m + 1 C (n, r): n = c x number of orientations of next piece

  10. Step 3 take 5x5 as an example Player 1

  11. Step 3 take 5x5 as an example Player 2

  12. Step 3 take 5x5 as an example There are other possible orders. Each player has to x (4!) Player 1: [C (1, 1) + C (8, 1) + C (14, 1) + C (40, 1)] x (4!) = 1512 Player 2: [C (24, 1) + C (8, 1) + C (14, 1) + C (40, 1)] x (4!) = 2064 1512 + 2064 = 3576 3576 >> 2500 Assume all the pieces offered can fit in the board; do not take the side-touch into consideration.

  13. Step 3 8x8 size: Player 1: [C (1, 1) + C (8, 1) + C (14, 1) + C (40, 1) + C (26, 1) + C (16, 1) + C (76, 1) + C (88, 1) + C (50, 1)] x (9!) = 115,758,720 Player 2: [C (63, 1) + C (8, 1) + C (14, 1) + C (40, 1) + C (26, 1) + C (16, 1) + C (76, 1) + C (88, 1) + C (50, 1)] x (9!) = 138,257,280 115,758,720 + 138,257,280 = 254,016,000 The total number of game states is 254,016,000.

  14. Step 3 14x14 size: Player 1: [C (1, 1) + C (8, 1) + C (14, 1) + C (40, 1) + C (26, 1) + C (16, 1) + C (76, 1) + C (88, 1) + C (50, 1) + C (56, 1) + C (124, 1) + C (136, 1) + C (148, 1) + C (160, 1) + C (86, 1) + C (184, 1) + C (196, 1) + C (208, 1) + C (220, 1) + C (58, 1) + C (244, 1)] x (21!) = 1.0928 x 10^23 Player 2: [C (195, 1) + C (8, 1) + C (14, 1) + C (40, 1) + C (26, 1) + C (16, 1) + C (76, 1) + C (88, 1) + C (50, 1) + C (56, 1) + C (124, 1) + C (136, 1) + C (148, 1) + C (160, 1) + C (86, 1) + C (184, 1) + C (196, 1) + C (208, 1) + C (220, 1) + C (58, 1) + C (244, 1)] x (21!) = 1.19195 x 10^23 1.0928 x 10^23 + 1.19195 x 10^23 = 2.28475 x 10^23 The total number of game states is 2.28475 x 10^23.

  15. Computer Program What can our computer program do? • 2 different versions. • Human only version • Computer only version.

  16. Move Input • Either player 1 or player 2 inputs the following: • Piece • Row • Column • Orientation • This will be demonstrated in a bit.

  17. Piece Representations • Since we did not use graphics in our program we represented the pieces as numbers. • You had to reference a sheet of paper that had the pieces drawn on them and all their orientations.

  18. Pieces Example

  19. Version 2 • This version of the program was the most helpful. • What does the program do? • Makes random legal moves. • Tell the computer know what size of board you want to use and how many games you want it to play. • Outputs the total number of different board configurations

  20. Some results • 3x3 board. • Most number of boards found was 574 • 5x5 board • Most number was 3445

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