# Connect 4 - Maths Investigation.

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Introduction

Connect 4 Introduction: I am going to investigate how many different ways you can win with connect 4. To start with I am going to use grids with a height of 4 and of varying lengths. From the table above I can see that there is a pattern with the results. The Length goes up by 4 every time, The Height goes up by 1, The Diagonal goes up by 2 and The Total goes up by 7. From this I can form the prediction that for 4 x 7 the Length will be 16, the Height will be 7, the Diagonal will be 8 and the Total will be 31. From the results I can see that my prediction was correct and I have now decided to work out a rule so that you can work out the total for any length with height 4. I am going to use the difference method to work out the rule. 4 X 4 4 X 5 4 X 6 4 X 7 Length 4 8 12 16 Height 4 5 6 7 Diagonal 2 4 6 8 Total 10 17 24 31 7 7 7 As you can see the difference is 7, using the difference method if ...read more.

Middle

From this I can see that my rule to work out how many ways to win connect 4 is correct. To Extend this course work even further I have decided to work out an over all rule so you can have any amount of connects, any height and any width. To start with I am going to look for connect 3's 3 x 3 3 x 4 3 x 5 Length 3 6 9 Height 3 4 5 Diagonal 2 4 6 Total 8 14 20 6 6 From the above table you can see that the difference between the totals is 6. Using the difference method to form an overall rule I multiply the difference (6) by the length and take away n to get the total for any connect. Premature rule: 6L - n For the 3 x 5: 6 x 5 - 10 Rule for Connect 3 with height 3: 6L - 10 Height 4 4 x 3 4 x 4 4 x 5 Length 4 8 12 Height 6 8 10 Diagonal 4 8 12 Total 14 24 34 10 10 From the above table you can see that the difference between the totals is 10. ...read more.

Conclusion

As height is the variable I will have to put it into the equation, I have decided to put it at the beginning. To work out the final rule I will have to put that into a current total. For Height5: [H(4L-12)] 5(4L-12) = 20L - 60 ~ to get the rule for height 5 you need to take away 12L and add 32. The final rule for Connect 3 is H(4L-12)-12L+32 Final Rule: To find the final rule I am going to use the difference method; the same that I used to find out all the other connect rules. Firstly I will have to tabulate all the rules so I can see the difference. Connect Rule 3 H (4L - 6) - 6L +8 4 H (4L - 9) - 9L + 18 5 H (4L - 12) - 12L + 32 So the final rule is: H[4L - 3(C-1)] - 3(C-1) L + 2(C-1)� Test Final Rule: To test the final rule I have decided to use a connect and grid size that I haven't used before. To test this rule I have decided to use connect 2 and a grid of length 2 and height 7. Using the rule substituting the letters for the appropriate latters. 7[4x2-3(2-1)]-3(2-1)x2+2(2-1)� = 35 From this I can see that my final rule is correct. ...read more.

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