The Integer Matrix

The Integer Matrix

Definitions
Object of the Game
Number of Players
Equipment
Rules
	(1) Setting the Players
	(2) Setting up the Board
	(3) Players Turn
		(A) Creating a Matrix
		(B) Adding the Matrix Score
		(C) Net Score Card
Winning the Game

Definitions

Integer – A member of the set of positive whole numbers {1, 2, 3, …}, negative whole numbers {-1, -2, -3, …}, and zero {0}.

Matrix - A rectangular array of numeric or algebraic quantities subject to mathematical operations.

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Object of the Game

The object of the game is to create a net positive or negative score that is farthest away from zero through the addition and subtraction of integer matrices.

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Number of Players

2 individual players, or 2 teams with 2 players on a team.

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Equipment

Game board
49 game discs
Game disc storage bag
Net score sheets

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Rules

(1) Setting The Players

Decide who is to be positive player and who is to be negative player in the game. The goal for the positive player is to accumulate a net score that is as far away from zero in the positive direction. The goal for the negative player is to accumulate a net score that is as far away from zero in the negative direction.

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(2) Setting Up The Board

The positive player reaches into the bag and removes a disc. Place it black side up on square 1 on the center of the game board. The negative player removes another disc from the bag and places it white side up on square 2 on the game board. Alternating positive player and negative player, continue placing pieces on the board until the 9 center squares are filled. (see Figure 1)

Figure 1: Game board set up



1	2	3
4	5	6
7	8	9

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(3) Players Turn

Negative player plays first.

Reach into the bag and remove a disc. Three things must then occur, the player: (A) creates a matrix on the board; (B) sums the matrix score; and (C) adds the matrix score to his or her net score card.

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*(A) Creating a Matrix*	Figure 2: Creating a Matrix
The player decides whether to place the disc white side up or black side up. The disc must be placed such that a row or column of discs is bordered at each end by the same color. Figure 2 shows an example where white disc A is already placed on the board. The placement of white disc B flanks the row of three black discs, and the black discs are turned over so they are white.

Figure 3 shows an example where flanking occurs on multiple lines, one vertical and one diagonal.

A disc must be placed such that a matrix is formed and at least one disk is turned over (black to white, or white to black).

Figure 3: Creating Multiple Matrices

Game Note: A player may not create a matrix that will cause the board to turn all one color. For example, if there are 8 white discs on the board and 1 black disc, the player may not place a disc that will cause the board to contain 10 white discs and 0 black discs. It must be played as a black disc, regardless of the scoring outcome.

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(B) Adding The Matrix Score

Each disc has two numbers, one side positive (or zero) and the other side negative (or zero). When the discs have all been flipped in a turn, the matrix of zeros, positive numbers, and negative numbers is added and the player receives that score as a matrix score.

Figure 4 shows the sum of a white matrix. Notice that the black disc (-4) is not included in the matrix score.

	Figure 4: Adding a matrix

Figure 5 shows the addition of multiple matrices (assuming the player placed the black –3 disc and created 2 matrices. Notice that the white discs are not included in the total matrix score.

Figure 5: Adding multiple matrices

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(C) Net Score Card

Figure 6: Using the Net Score Card

At the end of a players turn, the matrix score is added to the net score for the player. For example, if the negative player received a matrix score of –2 on his first turn, and scores of –3, +1, -11, and 0 on his subsequent turns, his Net Score Card would look like figure 6.

Thus, after 5 turns, the negative player would have a score of –15.

Play continues back and forth, negative player to positive player, until the game board is filled with 49 discs, or until it is no longer possible to place a disc such that another disc can be turned, creating a new matrix.

Net Score
-2
-3
-5
+1
-4
-11
-15
0
-15

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Winning The Game

At the completion of the game, the player with the score farthest from zero (greatest absolute value) is declared the winner. Refer to the example below.

Game A		Game B
Positive Player Score: +64		Positive Player Score: +44
Negative Player Score: -50		Negative Player Score: -78
Winner: Positive Player		Winner: Negative Player

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