• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13

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.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Numerical Method of Algebra.

    I only got its range, -0.66514 < root of the equation used < -0.66513. So, I decide to get their error bound by: Average of the upper and lower error bound = = Upper Error Bound = Lower Error Bound = The Error Bound of the root of this equation

  2. The open box problem

    X 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 V 12.5 13.824 14.812 15.488 15.876 16 15.884 15.552 15.028 14.336 13.5 After looking at this table I have concluded that the maximum volume for the open box dimensions of 6x6 is 16, and x (that makes the volume at it's maximum is)

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    244 18.87h October 1 274 17.95h November 1 305 17.12h December 1 335 16.65h Toronto Location: 44?N 79?W Date Day Number of Hours of Daylight January 1 1 8.95h February 1 32 9.86h March 1 60 11.19h April 1 91 12.75h May 1 121 14.17h June 1 152 15.25h July

  2. Triminoes Investigation

    to try out the formula by putting in the n th term, prove that it works. Substitute n=1 f(1) = 1/6 x 13 + 1 x 1� + 11/6 x 1 + 1 = 1/6 + 1 + 11/6 + 1 = 12/6 + 2 = 2 + 2 = 4 Substitute n=1 F(6)

  1. Experimentally calculating the wavelength of an He-Ne laser by means of diffraction gratings

    Also, the distances between the central beam and the first order fringe to the left and the first order fringe to the right seem to be equal. This is the same for the second and third order fringes. However, there seems to be slightly more space between the first order

  2. Maths - Investigate how many people can be carried in each type of vessel.

    We will begin by taking one of the equations that was formed earlier (equation (iv)) and will substitute y = 5 into the equation. y - z = 1 5 - z = 1 -z = -4 z = 4 Now we have already solved two variables of the equation,

  1. Functions Coursework - A2 Maths

    a root of the equation f(x)=0 lies in the interval [1.87,1.88]. The rest of the method could also be illustrated graphically in this way. The next table show values of f(x) at values of x from 1.87 to 1.88, with intervals of 0.001.

  2. Solutions of equations

    I will know find the values for the function between 0.6 and 0.7. Table 3 x F(x) 0.6 -0.392 0.61 -0.377 0.62 -0.349 0.63 -0.306 0.64 -0.247 0.65 -0.171 0.66 -0.075 0.67 0.0412 0.68 0.1797 0.69 0.3418 0.7 0.529 Table 4 x F(x)

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work