T-totals. For my T-totals maths coursework I will investigate the relationship between the T-total and T-number, the T-total and T-number and grid size and the T-shape in different positions.

                

This means that to find the T-total or T-number of a T-shape on a grid size 9 x 9 you must use this formula:

T = 5N – 63        (T = T-total)

5N = There are 5 numbers in a T-shape.

-63 = The total difference between all the numbers and the                            

 T-number.

Now we must put this formula to the test,

For example, if I put the T-number 41 into the formula it looks like this:

5 x 41 – 63 = T-total

205 – 63 = T-total

T-total = 142

Therefore, the T-total of the T-shape with a T-number of 41 on a 9 x 9 grid should be 142.

T-total = 22+23+25+32+41=142

Therefore my first formula was proven correct! √

This means that I can also find the T-number if I have the T-total of a 9 x 9 grid if I use the formula:

T-number = T + 63

                          5

Now we must put this formula to the test,

For example, if I put the T-total 217 into the formula it looks like this:

217 + 63  = N

      5

280 = N

  5

N = 56

Therefore, the T-number of the T-shape with a T-total of 217 should be 56.

I will now find the formula for a 6 x 6 grid:

By doing the same as in the 9 x 9 grid you will get a T-shape like this:

=

This means that the formula to find the T-number in a 6 x 6 grid is:

T = 5N – 42.

Now we must put this formula to the test,

For example, If I put the T-number 27 into the formula it looks like this:

5 x 27 – 42 = T

135 – 42 = T

T = 93

Therefore, the T-total of a T-shape with a T-number of 27 on a 6 x 6 grid should be 93.

 

T-total = 14+15+16+21+27=93

Therefore my formula was proven correct! √

Part 2

In the last part I noticed that not only did the centre column in each T-shape for the 9 x 9 grid go up in 9’s but also the centre columns in each T-shape for the 6 x 6 grid went up by 6. This means that I can work out a formula for any T-total on any grid size by using the calculations below:

                (G = Grid size)

This means that to find the T-number on any T-shape on any size graph you must use the formula:

T = 5N – 7G

Now I must put this formula to the test

I will use a 5 x 5 grid:

I will use the T-number 18.

T = 5N – 7G

T = 5 x 18 – 7 x 5

T = 90 – 35

T = 55

This means that the T-total for a T-shape with T-number 18 on a 5 x 5 grid should be 55.

 T = 7+8+9+13+18=55        

I have proven that it works on a square grid but now I will see if it will work on a rectangle grid. I will use a 4 x 6 grid and use the T-number 19.

T = 5N – 7G

T = 5 x 19 – 7 x 4

T = 95 – 28

T= 67

T = 10+11+12+15+19=67

This means that my formula is correct and now I can make a formula that gives me the T-number if I have the T-total. This formula is:

N = T + 7G

          5

Now I will test this formula using a 4 x 4 grid and using a

T-total of 42.

N = T + 7G

          5

N = 42 + 28

           5

N = 70

        5

N = 14

N = 14

My formula was proven correct!

Now I will try to find a relationship for the T-shape in different positions on any grid. Firstly, a 7 x 7 grid:

I will firstly use two T-shapes in “upside down” positions:

                 

        &

As with at the beginning when I had to first find a formula I have noticed a relationship between the T-number of each T-shape and the other numbers in its T-shape.

This relationship will enable me to find formulas for T-shapes in any positions on any grid:

This means that to find the T-total or T-number of an “upside down” T-shape on a grid size 7 x 7 you must use this formula:

 

T = 5N + 49

        Now we must check this. For example, if I use the T-number 33 into the formula it looks like this:

T = 5N + 49

T = 5 x 33 + 49

T = 165 + 49

T = 214

T = 33+40+46+47+48=214

Therefore my formula is correct.

Now I know that to find the formula of the T-shape “upside down” or “opposite” you must turn the mathematical signs opposite I can now find the formula for any size grid without doing much working out:

This means that the formula to find a T-total of an upside down T-shape on any size grid is:

T = 5N + 7G

Now I must test this formula. I will use a 8 x 8 grid and the T-number 28:

T = 5N + 7G

T = 5 x 28 + 7 x 8

T = 140 + 56

T = 196

T = 28+36+43+44+45=196

Therefore my formula is correct.