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Introduction

Relationship between T-total and T-number

I am going to investigate T-totals and T-numbers on a 9x9 grid.

The T-total is when a T-shape is placed on a grid and the numbers in the T-shape are added up to give a total.

My aim is to find a relationship between T-totals and T-numbers.

I am going to translate the T-shape 5 times on a 9x9 grid and then do the same on various sized grids.    T total = 1 + 2 + 3 + 11 + 20 = 37 [All numbers added together]  T number = 20 [number at the bottom of the T shape]

I am going to translate the T shape across the 9x9 grid.      T total = 1 + 2 + 3 + 11 + 20 = 37

T number = 20    T total = 2 + 3 + 4 + 12 + 21 = 42

T number = 21      T total = 3 + 4 + 5 + 13 + 22 = 47

T number = 22  I have translated the T shape across the 9x9 grid 3 times. So far I have noticed that when the T number increases by one, the T total increases by 5.

Middle

T total = 10 + 11 + 12 + 20 + 29 = 82

T number = 29      T total = 19 + 20 + 21 + 29 + 38 = 127

T number = 38

I have noticed that as the T number increases by 9, the T total increases by 45. I am going to translate two more T shapes onto the grid to test my prediction.      T total = 28 + 29 + 30 + 38 + 47 = 162

T number = 47 T total = 37 + 38 + 39 + 47 + 56 = 217   T number = 56  I have tested my prediction and I can see that it is correct. As the T number increases by 9, the T total increases by 45.

Across : Down : I have made a table, which shows the increase in the T total when the T number increases.

I have made a table for

Conclusion

3x3 grid T total = T + T-3 + T-7 + T-6 + T-5

= 5T – 21

CHECK: T total = 5 x 8 – 21 = 19

T = 1 + 2 + 3 + 5 + 8 = 19

I have found the formula for various sized grids. I have tested my formulas and they all work to find the T total.

I noticed that each time the grid size is decreased by 1, the number you take away from 5T increases. [Increases because it is a –minus]

I am now going to find a formula for all grid sizes.

## GENERALISING

I have found the formula for different grid sizes and have also found information for each grid. ## G stands for the grid size T – (2G-1)
= 20 – (2x9 –1)
= 3

T NUMBER

T – G
= 20 – 9
= 11

T-total= T+T-G+T-2G+T-2G+1+T-2G-1

=5T-7G

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

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