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T-Totals. To figure out an equation for different grid sizes, I have to find the relationship between grid sizes and the T total. I will now let S= Grid Size.

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Introduction

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T-Totals

Part 1

By using algebra I can find the equation for T Total for grid size 9

                                                       T Total= n-19+n-18+n-17+n-9+n= 5n-63

T Total= 5n-63 is the relationship between the T total and the T number. This means for example that if the T number is 20 my formula predicts a T Total of 5× 20– 63= 37 which agrees with my earlier calculations.

Part 2

Equation for different grid sizes

To figure out an equation for different grid sizes, I have to find the relationship between grid sizes and the T total. I will now let S= Grid Size.

               I get this T

T Total= n+n-S+n-2S+n-2S+1+n+2S-1= 5n-7S

This means that the equation is T total= 5n-7S where S is the grid size so if for example the T number is 24 and the grid size is 11 then the T total will be (5× 24– 11× 7)= 43. I can check whether this works.

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The totals are the same so my formula seems to work.

Translation

I will now try and figure out the relationship between translation and the T Total.

...read more.

Middle

79

The resulting T total is 22+23+24+32+41= 142.

Using my equation for a translated t total, New T total= T+5x+5(yS), for a T total of 37, a grid size of 9 and translation of image01.png then the equation will be  T=37+15+90= 142 which the same as the T total on the grid, so my formula works.

Another way of doing this is by representing the number in each square by a letter.

Again I will use image00.pngto represent the vector movement, S for the grid size T for the new T Total.

a+x+yS+ b+x+yS+ c+x+yS+ d+x+yS+ e+x+yS=

a+b+c+d+e+5x+5yS= New T Total

a+b+c+d+e is equal to the old T Total which I will call T so the formula is

T+5x+5yS= New T Total

This gives the same equation as in the previous result so I know works as I showed it in the previous result.

Rotation

I will now try and work out the relationship between the T number and the T total when it is rotated about the original T number.

n= T number

S= Grid size

T= New T total

I will try a 90º clockwise rotation

n+n+1+n+2+n+2-S+n+2+s= 5n+7

T=5n+7

This is the T for a 180º clockwise rotation      

n+n+S+n+2S-1+n+2S+n+2S+1=7n+7S

T= 7n+7S

This is the T for a 270º clockwise rotation

n+n-1+n-2+n-2-S+n-2+S= 7n-7

I can show that these formulas work by drawing a table.

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The formula for the 90º clockwise rotation is T= 5n+7 so the T number is 20 so the T total would be (5×20) +7= 107. From the table I can see that the T total is 25+26+27+17+37= 107 so my formula works.  

The formula for the 180º clockwise rotation is T= 7n+7S so the T number is 20 so the T total would be (7×20) + (7×10)= 210. From the table I can see that the T total is 25+35+45+46+44= 210 so my formula works.

The formula for the 270º rotation clockwise is T= 5n-7 so the T number is 20 so the T total would be (5×20) -7= 93. From the table I can see that the T total is 25+24+23+13+33= 107 so my formula works.  

I will now try and figure out the relationship between the T number and the T total in a 90º clockwise rotation about a point on the table.

T number= n

New T total= T

Grid Size= S

Horizontal distance from the T number

To the point of rotation= x

Vertical distance from the T number

To the point of rotation = y

= Distance from the point of rotation

= Rotation Point

           = The T number

To figure it out I would have to

I can prove this by drawing a grid

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...read more.

Conclusion

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I will now make a T for a rotation of 270º clockwise in the same format as the last one

n+yS+x+xS-y+n+yS+x+xS-y-1+n+yS+x+xS-y-2+n+yS+x+xS-y-2-S+ n+yS+x+xS-y-2+S=

5n+5yS+5x+5xS-5y-7= T

T= 5(n+yS+x+xS-y)-7

To show that it works, I made an example.

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Limits on T

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...read more.

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