* by inspection I can see this rule.
.. I used the difference method to find this rule:
3 4 5 6
2 8 18 32
\ / \ / \ /
6 10 14
\ / \ /
- 4
This shows that the rule will begin with 2n².
-8 -8 -8
D = number of diagonal lines.
B= D-2n²
This means my rule so far is 2n²-8n.
(n² x 2)-(8 x 3)= 18-24= -6
-6 + 8= 2 +8
(n² x 2)-(8 x 4)= 32-32= 0
0 +8= 8 +8
Therefore the rule for diagonal lines in an n x n grid is 2n²-8n+8.
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:
To find the rule I will use the difference method.
14 34 62 98
\ / \ / \ /
20 28 36
\ / \ /
- 8
Half of 8 is 4. So the rule will begin with 4n².
-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 going to confirm my rule by breaking down the winning lines into vertical, horizontal and diagonal, and then adding together the rules.
* by inspection I can see this rule.
.. I used the difference method to find this rule:
3 4 5 6
4 12 24 40
\ / \ / \ /
8 12 16
\ / \ /
- 4
This shows that the rule will begin with 2n².
-6 -6 -6
D = number of diagonal lines.
B= D-2n²
This means my rule so far is 2n²-6n.
(n² x 2)-(6 x 3)= 18-18= 0
0 + 4= 4 +4
(n² x 2)-(6 x 4)= 32-24= 8
8 +4= 12 +4
Therefore the rule for diagonal lines in an n x n+1 grid is 2n²-6n+4.
4n² - 8n + 2
I am now going to look at winning lines of 3 in a different n x m grid- where
m = n+2.
3 x 5 4 x 6 5 x 7 6 x 8
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:
To find the rule I will use the difference method.
20 44 76 116
\ / \ / \ /
24 32 40
\ / \ /
- 8
Half of 8 is 4. So the rule will begin with 4n².
-4 -4 -4
4n²- 4n
(n² x 4)-(4 x n)
(9 x 4)-(4 x 3)= 36-12= 24
24 - 4= 20 -4
(16 x 4)-(4 x 4)= 64-16= 48
48 - 4= 44 -4
Therefore the rule for n x m- where m= n+2 is: 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.
* 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
\ / \ /
- 4
This shows that the rule will begin with 2n².
-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.
