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Investigate the number of winning lines in the game of connect 4.

Extracts from this document...

Introduction

Investigation

Connect 4

X

X

X

X

Task

This is a winning line in the game of connect 4 on a 4x5 board. Winning lines can be horizontal, vertical and diagonal. Investigate the number of winning lines in the game of connect 4.

The task is asking me to find out how many winning lines (connects) when you are connecting 4 there are on any size board.

What am I going to do

I am going to find out how many connect 4 there are in a 4x5 board.

•I will change the size of the box, but keep one value the width constant. And I will find a pattern in the number of connects there are in the different size boxes.

•I will use algebra to find a general formula for a NxWidth (W) box.

•I will then increase the width (constant) by one and work out a formula for that box.

•I will then find a pattern in the formulas for the different size boxes, connecting 4, and I will make a formula for the formula.

•I will then change the number that I will connect. For example 2, 3 or 5.

Connect 4

Firstly I will do a box with the width constant as 5 and I will change the height.    

Hx5 Box

Any Number=N

Connects=C

Height= H

Width =W

                    Hx5   1   2   3   4    5    6        

            Connects   2   4   6  17  28  39          first layer

                                         11  11  11             second layer                                                            

The box height of 1 and 2 do follow the pattern so

they are excluded. The connects go up by 11 each

time.   There are only 2 layers so the equation we

use is this equation. C=aH+b (original equation)

...read more.

Middle

As before the first 2 equations do not follow  

the pattern so they are excluded. Also like before  

the connects has 2 layers so we use this original

equation.

C=aH+b

We use the same method as before.

  9=a3+b     (1)

24=a4+b     (2)

15=a           (2)-(1)

Substitute ‘a’ which is 15 back into (1)

9=15x3+b

9=45+b

                           b= -36

Substitute‘a’ and ‘b’ back into original equation

C=15H-36    that is the equation for the number  

of connects in a Nx6 box. But since the            

          first 2 heights didn’t follow the        

          pattern we didn’t use them in the    

          equation so this equation doesn’t      

                                           work for them.

Connect 4

Hx4 Box

         Hx4   1   2   3   4   5   6

        Connects   1   2   3  10 17 24     first layer

                                      7   7   7         second layer

                                  Like before the first to equations do not follow the    

           pattern so they are excluded. Also like before the        

           connects has 2 layers so we use this original

           equation.

                       C=aH+b

We use the same method as before.

  3=a3+b     (1)

10=a4+b     (2)

7=a             (2)-(1)

Substitute ‘a’ back into (1)

3=7x3+b

3=21+b

b= -18

Substitute ‘a’ and ‘b’ back into original equation

C=7H-18  

that is the equation for the number of connects in a Nx4    

box. But since the first 2 heights didn’t follow the

pattern we didn’t use them in the equation so this        

   equation doesn’t work for them.

Formula For Connect 4

The formula for any box with a width of 4 is C=7H-18

The formula for any box with a width of 5 is C=11H-27

The formula for any box with a width of 6 is C=15H-36

...read more.

Conclusion

(1)

18=a16+b4+z   (2)

32=a25+b5+z   (3)

14=a9+b        (3)-(2)       (4)

10=a7+b        (2)-(1)       (5)

  4=2a            (4)-(5)

  2=a

Substitute ‘a’ back into (4)

14=2x9+b

14=18+b

b=-4

Substitute ‘a’ and ‘b’ into (3)

32=2x25-4x5+z

32=50-20+z

32=30+z

z=2

So we now know what ‘a’, ‘b’ and ‘z’ are so we sub them back into the original equation which was Fo=aC²+bC+z  

Fo=2C²-4C+2 is the fourth number equation

So the first number equation is 4

The second number equation is 3C-3

The third number equation is 3C-3

The fourth number equation is Fo=2C²-4C+2

So we now join them together.

In the connect number equations the first 2 numbers were in brackets and so were the second 2. So we have to group the first 2 equations and the second 2 in brackets. But each equation has to be in its own brackets so we need to use double brackets.

The formula for connect 3 was (4W-6)H-(6W-8) we will use it as a bass.

(first number equation xW-second number equation)H-(third number equation-Fourth number equation)  

But each number equation needs to be surrounded by its own brackets.          

((first number equation xW)-(second number equation))H-((third number equation)xW-(Fourth number equation))

((4W)-(3C-3))H-((3C-3)xW-(2C²-4C+2))

This formula finds out how many connects there are in any size box using any connecting number. E.g you could use a height of 5 and width of 4 and we are connecting 4. This would give you the answer of 17 which is correct.

But as before the formula does not work if the height is 2 or more numbers lower than the number you are connecting.

...read more.

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