# T- Total. When trying to fine out the T-Total in a 6 by 6-number grid, If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like the one shown below the 6x6 grid.

Extracts from this document...

Introduction

T- Total

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 |

Look at the T-shape drawn on the 9 by 9 number grid.

The numbers in this shape are added together to make the T-Total, which would be in this case 1+2+3+11+20=37. So the T-Totals number would be 37.

As I go on I am sure I will find a pattern which may help me find a rule.

## T-Totals | ## T-Number |

2+3+4+12+21=42 | 21 |

3+4+5+13+22=47 | 22 |

4+5+6+14+23=52 | 23 |

The number at the bottom of the T-shape is called the T-number.

The T-number for this T-shape is 20.

If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.

T-17 | T-18 | T-19 |

T-9 | ||

T |

You will notice that the centre column of the T-Shape is going up in 9’s this is because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this a 9x9 grid. This is done in the working below: -

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

= 5T-63

Now I will test this formula to see if it works for the 9x9 Grid

### For T-Total I will use the letter X

For the T-Number I will use the letter T

So X = 5T-63

T = 20

X = 5x20-63

= 100-63

= 37

Middle

T

This time the centre column of the T-Shape is going up in 6’s because of the table size. The formula for this size grid is T-Total = T-13+T-12+T-11+T-6+T= 5T-42

Now to do as I did before I will test my formula out: -

With X = T-Totals and T= T-Number

T = 35

So X= 5x35 – 42

This means X should equal 133

The numbers in the T-Shape are 22,23,24,29,35 and if you add them all together this makes 133 which shows my formula is true, but just to make sure I will test it again.

### T = 16

So X = 5x16 – 42

So X= 32

3 + 4 + 5 + 10 + 16 = 32

### This shows that my formula is correct!

I have tested the formula for two different T-Numbers and the formula works on a 6x6 grid.

The full formula for this size grid is: - X=5T-42

Below is a grid with parts of the formula in. I will use this to figure out a formula for any T-Total on any width grid.

T-2G-1 | T-2G | T-2G+1 |

T-G | ||

T |

T-total = T-2G-1

T-2G

T-2G+1

T-G

### T

X=5T-7G

If the Grid Width (= G), and the T-Number (= T), the T-Total (= X) is calculated by

X=5T-7G

Now I’m going to try the grid that I drew earlier which was G=9

The formula is X=5T-7G

Which changes to X=5T-7x9

X=5T-63, this shows that this rule has a relationship between itself and the rule for the grid.

I will now try the formula on a 7 by 7 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 |

X=5X17-49(2+3+4+10+17=36) – This is what the numbers in the T-Shape added together equals.

### X=85-49

X=36

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

10 | 11 | 12 |

13 | 14 | 15 |

I am now going to try this formula on a 3 by 5 grid.

X=5X8-21 (1+2+3+5+8=19) – This is what the numbers in T-Shape added together equals.

### X=40-21

X=19

I am now going to try this formula on a 6 by 10 grid.

X=5X33-42 (20+21+22+27+33=123

### X=165-42

### X=123

Now I have worked the formula out for the T-Shape in the one position, I am going to translate the T-Shape to different positions and investigate the relationships between the T-Total, the T-number and the grid size.

As you can see below the T-Shape can be set out just like the previous one I drew but the formulas are flipped over.

## T | ||

T+G | ||

T+2G+1 | T+2G | T+2G-1 |

T + (T+G) + (T+2G+1) + (T+2G) + (T+2G-1) = 5T + 7G

#### As you will see the rule is fairly similar but instead of a take-away sum it is an additional sum, I think this happens because usually when you flip something it is the opposite to what it was and when you flip the T-Shape it flips the rule.

I will now try this new formula on a 4 by 7 grid, using 10 as the T-Number.

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 |

Conclusion

=73

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

### I have tried this new formula for an inverted T and it works, the formula is: X=+5T

### I will now find out a formula that has been translated 90 degrees anti-clockwise

T-1-G | ||

T-2 | T-1 | T |

T-2 + G |

T

### T-1

T-2

### T-2-G

### T-2+G

### 5T-7 this is what all the squares added above equals and this should be the rule.

### I will use this rule on a 5by5grid this should not affect any outcome of the end result, I will use 8 as my T-Number though

X= 5T-7=5x8-7

### X= 40-7=33 (Numbers In T-Shape = 1+6+11+7+8= 33)

### So my rule is correct!

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 |

I have tested this new formula for a side ways T the formula works it is: X=5T-7

###### Before I write the formula I will predict what it will be 5T+7

T+2-G | ||

T | T+1 | T+2 |

T+2+G |

T

T+1

T+2

T+2+G

T+2-G

5t+7

The formula is 5T+7 My Prediction was correct!

I have tested this new formula for a side ways T the formula works it is: 5T+7

The formula for a this way up is: X=5T-7G

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

The formula for a this way up is: X=7G+5T

## 1 | ## 2 | ## 3 | ## 4 |

## 5 | ## 6 | ## 7 | ## 8 |

## 9 | ## 10 | ## 11 | ## 12 |

## 13 | ## 14 | ## 15 | ## 16 |

### The formula for a this way up is: X=5T+7

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

The formula for this way up is: X=5T-7

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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