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Introduction

T-Total

## Part 1

The aim of the investigation is to find out the relationship between the t-number and t-total. The t-number is the number in the t-shape, which is at the base of the T. The t-total is the sum of all numbers inside the t-shape.

I will start my investigation by looking at t-shapes on a 9 by 9 grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

To solve the problem of finding the relationship between the t-number and t-total I will look at the information algebraically.

I will firstly assign a letter to the t-number of the shape, this letter will be T. I will then express the rest of

Middle

37

38

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41

42

43

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45

46

47

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53

54

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63

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81

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

The expression works for all t-shapes in a 9 by 9 grid. I will now simplify the expression into a simple formula.

T + (T-9) + (T-17) + (T-18) + (T-19) = T-total

## 5T – 63 = T-total

I will see if this new formula still works.

5 × 20 – 63 = T- total

100 – 63 =T – total

37 = 37

5 × 21 -  63 = T- total

105 – 63 = T – total

42 = 42

## Part 2

I will now as part of my investigation use different grid sizes, transformations of the t-shape and investigate the relationship between both. Then I will see how the t-number and the t-total relate to the new factors.

The smallest grid size can only be a 3 by 3 grid because that is the smallest grid size a t-shape can fit on. However this size grid will not allow me to translate the t-shape therefore I will start the investigation by using a 4 by 4 grid. Also I will be keeping the grid sizes square, so that the length of the grid is the same as its width. The largest grid size I can use is infinite.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

I will now look at a 5 by 5 grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

I will now look at a 6 by 6 grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

I will now look at the data algebraically that I have collected from each of the grids. I will be using the letter T as before to represent the t-number and then express the rest of the numbers in the t-shape with the letter T. This should give me a formula as before which works for each of the t-shapes from each of the grids.

## Data from 4 by 4 grid

10 + 6 + 3 + 2 + 1 = 22

T + (T-4) + (T-7) + (T-8) + (T-9) = T-total

Does this formula work for work for other t-shapes in a 4 by 4 grid.

6 + 7 + 8 + 11 + 15 = 47

T + (T-4) + (T-7) + (T-8) + (T-9) = T-total

Yes the formula does I will now simplify the formula.

T + (T-4) + (T-7) + (T-8) + (T-9) = T-total

5T – 28 = T-total

## Data from 5 by 5 grid

1 + 2 + 3 + 7 + 12 = 25

T + (T-5) + (T-9) + (T-10) + (T-11) = T-total

Does this formula work for work for other t-shapes in a 5 by 5 grid.

3 + 4 + 5 + 9 + 14 = 35

T + (T-5) + (T-9) + (T-10) + (T-11) = T-total

Yes the formula does I will now simplify the formula.

T + (T-5) + (T-9) + (T-10) + (T-11) = T-total

## Data from 6 by 6 grid

1 + 2 + 3 + 8 + 14  = 28

T + (T-6) + (T-11) + (T-12) + (T-13) = T-total

Does this formula work for work for other t-shapes in a 6 by 6 grid.

2 + 3 + 4 + 9 + 15  = 33

T + (T-6) + (T-11) + (T-12) + (T-13) = T-total

Yes the formula does I will now simplify the formula.

T + (T-6) + (T-11) + (T-12) + (T-13) = T-total

5T – 42 = T-total

I will now look at all the formulas from each of the grids and see how they relate to each other and see if I can work out another formula, which will allow me to calculate the t-total whatever grid size, it may be.

Formula

Grid size

5T – 28 = T-total

3 by 3

5T – 35 = T-total

4 by 4

5T – 42 = T-total

5 by 5

## 5T – 63 = T-total

9 by 9

The number, which you minus from the formula in all of the formulas are a multiple of seven. If you divide this figure by seven you get the grid size.

28 ÷ 7 = 4        42 ÷ 7 = 6

35 ÷ 7 = 5        63 ÷ 7 = 9

If we input this information into the formula we end up with an overall formula to work out the t-total by only knowing the t-number.

T-total = 5T – (7 ×Grid size)

## T-total = 5T – 7G

I will now see if this formula works for any grid size.

 24 25 26 34 43

Conclusion

5 by 5

 Formula Grid size 5T – 28 = T-total 3 by 3 5T – 35 = T-total 4 by 4 5T – 42 = T-total 5 by 5

Also the is a similarity with the value added to the formulas The formulas like the original t-shape formulas differ form grid size to grid size and again the values added are the same as the original t-shape formulas. So the values added are a multiple of seven when dividing the values added to the formula you get the size of the grid.

28 ÷ 7 = 4        42 ÷ 7 = 6

35 ÷ 7 = 5        63 ÷ 7 = 9

If we input this information into the formula we end up with an overall formula to work out the t-total by only knowing the t-number.

T-total = 5T + (7 ×Grid size)

## T-total = 5T + 7G

I will now see if this formula works for any grid size.

 15 24 32 33 34

## 5T + 7G = T-total

5 × 15 + 7 × 9 = T-total  60 + 63 = T-total  138 = T-total  183 = 138

 6 14 21 22 23

## 5T + 7G = T-total

5 × 6 + 7 × 8 = T-total  30 + 56 = T-total  86 = T-total  86 = 86

 22 32 41 42 43

## 5T + 7G = T-total

5 × 22 + 7 × 10 = T-total  110 + 70 = T-total  180 = T-total  180 = 180

The formula works for other grid sizes as well.

5T – 7G = T-total ⇒ Overall formula for t-shape without any rotation.

## 5T + 7G = T-total ⇒ Overall formula for t-shape rotated 180°.

The formulas with a negative rotation and no rotation at all have a value, which is subtracted in the formula. All the formulas with a positive rotation and a rotation of 180° have a value in the formula, which is added.

Each of the formulas have 7 and 5 as fixed values, only two of the formulas have the grid size involved in their formulas. This is because the t-totals differ greatly from grid size to grid size for t-totals, which come from t-shapes with a rotation of 180° and t-shapes without any rotation.

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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