i.e. L=W=T
3x3
8 winning lines
4x4
10 winning lines
5x5
12 winning lines
I got this rule because you can see there is a gap of two between each number in the number of winning lines column so this must mean the rule must be something times 2. I am using the width in this case so I times that, so I no the first part of the formula is 2W. To find out the rest of the formula I simple times each of the widths by two and see what the difference is between that and the number of winning lines. I then add this number to the 2W I already have. When this is done I have my rule.
To make sure this general rule is correct i will use the rule to find out how many winning lines would be in a 10x10 grid.
From my rule a 10x10 should have 22 winning lines
2x10+2=22
I will now draw out the grid for 10x10 to check my rule is correct
10x10
22 winning lines. My rule is correct
My rule is correct because
Investigation 2
Next I am going to investigate square grids using only 3 squares as a winning line.
i.e. W=L = T, T=3
3x3
8 winning lines
4x4
24 winning lines
5x5
48 winning lines
To check if my rule is correct I wall use it to tell me how many winning lines would be in a 10x10 grid. I will then check this by drawing out the grid. If my rule is correct there should be 288 winning lines.
10x10
288 winning lines my rule is correct
Overall 4W²-(12W-8)
=4W²-12W+8
=4(W²-3W+2)
=4 (W-1) (W-2)
In this case T=3 so I will write all number in my formula in relation to T in order to try and find a general formula for all squares and link my new formula to my first formula of 2W+2.
= (T+1) [W- (T-2)] [W- (T-1)]
=(T+1) (W-T+2) (W-T+1)
In investigation one I found the general formula 2W+2
My new formula is (T+1) (W-T+2) (W-T+1) which I found from investigation two
However, in investigation one T=W=L. Therefore I can relate my new formula back to investigation one by replacing T by W
= (W+1) (W-W+2) (W-W+1)
=(W+1) x 2 x 1
=2 (W+1)
=2W+1
So the formula from investigation one is a simplistic version of the formula from investigation 2
Therefore I now have a general formula for when the grid is a square and the length of the winning line is a variable
Investigation 3
For the next part of this investigation I am going to change the shape of the grid from a square to a rectangle. I will keep the length of the rectangle 3 and only change the width of the rectangle. I will use 3 as the number of squares to get a winning line.
3x4
14 winning lines
3x5
20 winning lines
3x6
26 winning lines
The rule for this is 6L-10. I can make sure this is the right answer by using it to tell me how many winning lines there would be in a 3x10 grid.
Using my rule in a 3x10 grid there would be 50 winning lines. To make sure this is defiantly correct I now need to check it by drawing out the 3x10 gird.
3x10
50 winning lines
10x6-10=50
My rule is correct
Investigation 4
Now I will investigate rectangles with a length of 4 and different variety of widths with 4 being the amount needed for a winning line.
4x5
17 winning lines
4x6
24 winning lines
31 winning lines
I will now use this rule to predict a 4x10
7x10=70-18=52
If my rule is correct in a 4x10 should contain 52 winning lines.
7x10
52 winning lines
My rules is correct
Conclusion
From this investigation I have found out a general rule for square grids when the length of the winning line is a variable. I would of found a general rule from the rectangle grids but from the patterns I choose it made it very hard to come up with one. If I were to extend this investigation further I would look more into rectangle patterns and come up with a general, I would also maybe look at some 3D grids and find out some rules from them. If I managed to find out all these rules I would work out a big general rule for the winning lines of a grid with any width, length or height if 3D.
