Duo By: Fernando & Vivian

GAME DESCRIPTION a set of ominoes with 21 Blokus tiles: Valid move: 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

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

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

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.

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

Step 1 Ominoes offered Free polyominoes of square one through N # of pieces of ominoes Total # of squares on the ominoes Corresponding size of board N = 2 2 3 3x3 N = 3 4 9 5x5 N = 4 29 8x8 All 21 pieces N = 5 21 89 14x14

Game states for smaller board 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;

combination for Bigger board 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

Step 3 take 5x5 as an example Player 1

Step 3 take 5x5 as an example Player 2

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.

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.

Step 3 14x14 size: The total number of game states is 2.28475 x 10^23. 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.

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

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

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.

Pieces Example

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

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