T-Totals. Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

 

   

N being T-number

I will now test my Rule (5n-63) to see if it is correct: -

I will take T-68 if I substitute this into my rule I will get (this is still on a 9*9 grid) 5*68-63=277

Sure enough the T-total is 68+59+51+50+49 = 277. This mean that my rule for standard upright T’s on a 9*9 grid is correct. Now for a upright T on any sized grid.

I will now investigate with using different grid sizes. If I now place the T-shape in the same place on the grid as I did at the start of this investigation you can see that the T-number has risen by 4 and the T-total has risen by six.

From this table I can see that the difference between the T-total is still 5 so the first bit of the formula is the same. Now to save time I will use a shorter method to work out the total difference between the T-totals and all the other numbers in the shape. in the last formula this number was 63.

24-1=23

24-2=22

24-3=21

24-13=11

TOTAL=77

Know I presume that this number is the variable that changes when the grid size does. I now have to change the formula to allow the grid size to be times or added to another number to reach my difference of 77. So I did this and obtained the formula, g being grid number.

5t-(7g)

I will now try this on another grid size to test my formula.

 

With my formula our will work out the T-total for T-number 22 on a 6*6 grid

5*22-(7*6)

=68

My formula has been proven to work on grids of a smaller size. I have now workout how to work out any upright t on any size grid just by working out how to obtain the last number in the formula.

I will now rotate the T-shape on its sun and see what part of the formula changes. I will then try to work out a formula for all rotations.

 I have to conclude that my formula will be changed in some way. There is no numbers change so I will try and change the minus sign to a plus and I will see if that works. The formula will be

5t+63 or 5t+(7*g)

I will now test this formula on another t-shape and on a different grid.

The T- number of this shape is 9 therefore

5*9+(7*6)

T-total=87

9+15+20+21+22=87

My formula for upside down T-shapes has now been proven to work. I will now turn the T-shape anti clockwise on its side and then the left. Then I will work out the formula.

Because the shape has not rotated a full 180 the sign at the end of the formula stays as a minus sign. To work out what I need to minus I will work out the difference.  I am going to do this using the long method just I case the time by 7 rule for the grid number does not work.

12-11=1

12-10=2

12-19=-7

12-1=11

Total=7

Therefore

5*12-7=53

I will now check if this formula works

T-number=12

12+11+10+19+1=53

My formula has been proven to work.

If I now rotate the shape through 180 degrees. I predict that the formula will be the same ecept I now will change the minus in the formula to a plus.

I will now test this theory

So the formula I will use is

5*10+7=57

T-number=10

10+11+12+3+21=57

This formula has been proven to be correct

I will now try to work a formula that I can use with vectors to work out the T-number and then use the formulas that I have already obtained to work out the rest.

This shape has been translated by a vector of  +4.

                                           -3

Now if my original T-number is 20 and I have moved I across right by 4 for every one square across I have added one to my original t- number. The same is for the other vector but for every one square up or down the grid number is added or subtracted.

So in this case

X- along in 1’s

Y- up r down in g

Therefore

(t+x)+(yG)

I will now test this

 

This T-shape has been moved by vector +3

                                        -4

So if I use my formula

Red T-number=32

(32+3)+(4*9)=71

Which is the Blue T-number now t work out the T-total all I have to do I use my formula from the first section.