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Noughts and Crosses Problem Statement:Find the winning lines of 3 in grids of n x n.

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Introduction

Noughts and Crosses

 Problem Statement:

Find the winning lines of 3 in grids of n x n.

    3 x 3                          4 x 4                      5 x 5                                 6 x 6        

Possible Solution:

To work out how many winning lines of 3 there are on each grid I will draw in the lines (see appendix). Here are my results:

3

4

5

6

8

24

48

80

To find the rule I will use the difference method.

3

4

5

6

8

24

48

80

\

/         \

/         \

/

16           24           32

                \  /             \  /

  1. 8

Half of 8 is 4. So the rule will begin with 4n².

n

3

4

5

6

a

8

24

48

80

4n²

36

64

100

144

B

-28

-40

-52

-64

\

/         \

/         \

/

                        -12         -12         -12

a = answer, as in number of winning lines.

B = a - 4n²

This gives my rule so far as 4n²-12n.

 (n² x 4)-(12 x n)

(9 x 4)-(12 x 3) = 36-36 = 0

0 + 8 = 8    +8

(16 x 4)-(12 x 4) = 64-48= 16

16+8= 24   +8

Therefore my rule is:  4n² - 12n + 8

To confirm my rule I will use an alternative method. I will break down “a” into the vertical, horizontal, and diagonal lines, find the rule for each then find the sum of the resulting rules.

...read more.

Middle

4n² - 12n + 8

This rule (the sum of the rules for vertical, horizontal and diagonal lines) is the same as the rule I found using the difference method, therefore it confirms that my rule is correct.

I am now going to look at grids of n x m.

In this case m = n+1

     3 x 4                 4 x 5                       5 x 6                               6 x 7

Possible solution:

To work out how many winning lines of 3 there are on each grid I will draw in the lines (see appendix). Here are my results:

3

4

5

6

14

34

62

98

To find the rule I will use the difference method.

14    34    62    98

    \  /    \  /    \  /

    20    28     36

        \ /      \ /

  1. 8

Half of 8 is 4. So the rule will begin with 4n².

n

3

4

5

6

a

14

34

62

98

4n²

36

64

100

144

B

-22

-30

-38

-46

\

/         \

/         \

/

                         -8           -8            -8

4n²- 8n

(n² x 4)-(8 x n)

(9 x 4)-(8 x 3)= 36-24= 12

12 + 2= 14     +2

(16 x 4)-(8 x 4)= 64-32= 32

32 + 2= 34     +2

Therefore the rule for n x m- where m= n+1 is: 4n²- 8n +2.

Again I am

...read more.

Conclusion

4n²- 4n -4.

Again I am going to confirm my rule by breaking down the winning lines into vertical, horizontal and diagonal, and then adding together the rules.

a

3

4

5

6

Vertical

9

16

25

36

n² *

Horizontal

5

12

21

32

n²-4 *

Diagonal

6

16

30

48

2n²- 4n

Total

20

44

76

116

4n² - 4n -4

* by inspection I can see this rule.

.. I used the difference method to find this rule:

3     4    5     6

6   16   30   48

   \ /   \ /    \ /

  10   14   18

      \ /    \ /

  1.  4

This shows that the rule will begin with 2n².

n

3

4

5

6

D

6

16

30

48

2n²

18

32

50

72

B

-12

-16

-20

-24

\

/                      \

/                      \

/

                                                     -4                        -4                          -4

D = number of diagonal lines.

B= D-2n²

This means my rule so far is 2n²-4n.

(n² x 2)-(4 x 3)= 18-12= 6

(n² x 2)-(4 x 4)= 32-16= 16

Therefore the rule for diagonal lines in an n x n+2 grid is 2n²-4n.

4n² - 4n -4

This rule (the sum of the rules for vertical, horizontal and diagonal lines) is the same as the rule I found using the difference method, therefore it confirms that my rule is correct.

I am now going to find a rule for any n x m grid.

...read more.

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