Investigate the relationship between the T-total and the T-number

This is correct because 58 + 59 + 60 + 68 + 77 = 322

Part 2 – Use grids of different sizes.  Translate the T-shape to different positions.  Investigate relationships between the T-total, the T-numbers and the grid size.

The difference between the T-totals is:

40             45                     50                    55               60                65                   70

      +5                 +5              +5             +5                 +5                 +5

I am going to see if the formula that I calculated earlier works for a 10 by 10 grid.

Therefore:  T = n + (n-21) + (n – 20) + (n – 19) + (n – 10)

To simplify: T = 5n – 70

This formula is similar except the number that is being subtracted.  However I will see if this works with a T-shape in the 10 by 10 grid.  

T = 5 x 26 – 70

T = 60

This is true because 5 + 6 + 7 + 16 +26 = 60

The equation is correct.  

However, in order to find the relationship between the grid size and the T-total and T-number it must be included in the equation.

So, when n = T-number

             T = T-total

             W = Grid size

This simplifies to: 5n – 7w

If the grid decreases to 6 by 6, the T-numbers will be closer together.  I noticed that on the previous 2 grids, the numbers being subtracted can be divided by the grid size and both equal –7.

 

I think that a 6 by 6 grid should be:

T = 5n – 42        (-7 x 6 = 42)

Part 3:  Use grids of different sizes again.  Try other transformations and combinations of transformations.  Investigate relationships between the T-total, the T-numbers, the grid size and the transformations.

REFLECTION

Horizontal reflection

38   ,    43                      39  ,  44                    40  ,  45

      +5                                +5                             + 5

The difference between the numbers in the T-shapes that have been reflected is 5.

The distance from the line of reflection and the T-number is 2, so reflected 2 x 2 = 4, but this is not a full reflection, so 1 must be added.  Therefore it would be n+5.

(n+5) + (n-w+5) + (n-2w+5) + (n-2w-1+5) + (n-2w+1 + 5)

T = 5n – 7w + 25

However, when this formula was tested with a different distance from the line of reflection it did not work.  This is because when tested 1 square from the line of reflection it should have been + 15 on the end of the equation instead of +25.

There is also an easier way to calculate the distance from the reflection line by using the letter a.

Here a = the distance from the line of reflection

(n+2a+1) + (n-w+2a+1) + (n-2w+2a+1) + (n-2w-1+2a+1) + (n-2w+1+2a+1)

This simplifies to: 5n – 7w + 10a + 5

I will now test this.

Using formula just calculated

(5 x 21) + (9 x -7) + (10 x 2) + 5 = T

T = 105 – 63 + 25

T = 67

This formula works with a T-shape that is 2 squares from the line of reflection, I will now see if it works when a = 3.  However, to do this it must be competed on a 10 by 10 grid because a 9 by 9 grid is too small.  This will also test if the formula works on different size grids.

Using formula just calculated

(5 x 62) + (10 x -7) + (10 x 3) + 5 = T

T = 310 – 70 + 35

T = 275  

This therefore proves that this formula is correct as it works on a 10 by 10 grid as well.

Vertical reflection

Here, b is the vertical distance from the line of reflection.  

        

   

With the previous formula the numbers only increased by one, whereas with a vertical reflection, the numbers are increasing by the grid size.  Also the numbers are in ascending order; here the T-number is the smallest number, whereas before it was the largest number.  

Therefore, because the numbers are in ascending order, the figures are being added instead of being subtracted.  Here, “b” causes the T-shape to be flipped; it is the movement of the T-number by double the distance from the line.

(n+2b+w) + (n+2b+2w) + (n+2b+3w) + (n+2b+3w+1) + (n+2b+3w-1)

This simplifies to: 5n + 10bw + 12w

I will now see if this formula works.

(25 x 5) + (10 x 2 x 10) + (12 x 10)

125 + 200 + 120 = 445

T = 445

This is true because when adding the numbers in the T-shape you get: 75 + 85 + 95 + 94 + 96 = 445

The formula is therefore correct and also works on a 10 by 10 grid.

TRANSLATION

When the T-shape with the T-number 75 is translated by the new T-number is 50.  The original T-total was:

 56 + 57 + 58 + 66 + 76 = 313 and is now 187.

In order to find a relationship and a formula I think it will be easier to find the skeleton of the T-number when it is being translated.

75 + 2 = 77

Then because the squares go up and down by grid size, I will multiply the grid size by the translation.

3 x 9 = 27

If this is subtracted from 77 it equals the new T-number.

If n = T-number

   W = Grid size

   Z = across translation

   U – Up or down translation

   N = new T-number

n + z – wu = N  

This means the remainder of the skeleton is based around this.  The other numbers were calculated by subtracting w and 2w, because above the T-number, the numbers decrease by the grid size for each row.    

(n+z-wu) + (n+z-wu-w) + (n+z-wu-2w) + (n+z-wu-2w-1) + (n+z-wu-2w+1)

This simplifies to: 5n + 5z – 5wu – 7w

I will now test this.  I will translate a T-shape with T-number 42 by

(5 x 42) + (5 x -4) – (5 x 9 x -2) – (7 x 9)

210 +  -20 - -90 – 63 = T

T = 210 – 20 + 90 – 63

T = 217

This is true because the T-total is 217 for this T-shape.  I deliberately used a translation with negative numbers to check that this did not alter the equation.

COMBINING REFLECTION AND TRANSLATION

n = T-number

W = Grid size

Z = across translation

U – Up or down translation

N = new T-number

a = horizontal distance from reflection line

b = vertical distance from the reflection line

T = T-total

HORIZONTAL REFLECTION TO TRANSLATION:

  5n – 7w + 10a + 5 + 5N + 5z – 5wu – 7w = T

The new T-number needed to be included in this equation otherwise it would not work, because the translation would occur from the original T-number.

This formula works because when tested on a 9 by 9 grid with the T-number being 21 and a being 2, the new t-number is 26 and when translated by 2 and -2 the T-total equals 147.  This agrees with the equation.

TRANSLATION TO VERTICLE REFLECTION:

5n + 5z – 5wu – 7w + 5N + 10bw + 12w = T

The new T-number needed to be included in this equation otherwise it would not work, because the reflection would occur from the original T-number.

This formula works because when tested on a 9 by 9 grid with the T-number being 76 and the translation being +2 by +6, the new T-number is 24.  When “b” is 1, the T-total equals 318.  This agrees with the equation.