T total = 4 + 5 + 6 + 14 + 23 = 52
T number = 23
T total = 5 + 6 + 7 + 15 + 24 = 57
T number = 24
I have tested my prediction and have found that my prediction was right. Now I am going to see what the relationship is between the T total and T number when I translate the shame down the grid.
T total = 1 + 2 + 3 + 11 + 20 = 37
T number = 20
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 when the shape is translated across the grid, and a table for when the shape is translated down the grid.
Across:
37 42 47 52 57
20 21 22 23 24
Down:
37 82 127 162 217
20 29 38 47 56
I am going to find a formula that links the T total and T number.
Total = T + T-9 + T-19 + T-18 + T-17
= 5T – 63
[T stands for the T number]
I have found a formula for translating the T shape on the 9x9 grid. I am now going to test my formula.
T total = 5 x 20 – 63 = 37
T = 1 + 2 + 3 + 11 + 20 = 37
- The above diagram and calculation verifies that my formula is correct and applies to all 9 by 9 grids.
I am now going to look at different grid sizes. I am going to start looking at it using algebra first, and then I will check it using real numbers.
8x8 grid
Total = T + T-8 + T-17 + T-16 + T-15
= 5T – 56
CHECK:
T total = 5 x 18 – 56 = 34
T = 1 + 2 + 3 + 10 + 18 = 34
The formula for the 8x8 grid is 5T-56. I am now going to find the formula for the 7x7 grid.
7 x 7 grid
Total = T + T-7 + T-15 + T-14 + T-13
= 5T – 49
CHECK:
T total = 5 x 17 – 49 = 36
T = 2 + 3 + 4 + 10 + 17 = 36
The formula for the 7x7 grid is 5T-49. I am now going to find the formula for the 6x6 grid.
6x6 grid
Total = T + T-6 + T-13 + T-12 + T-11
= 5T – 42
CHECK:
T total = 5 x 14 – 42 = 28
T = 1 + 2 + 3 + 8 + 14 = 28
The formula for the 6x6 grid is 5T-42. I am now going to find the formula for the 5x5 grid.
5x5 grid
T total = T + T-5 + T-11 + T-10 + T-9
= 5T – 35
CHECK:
T total = 5 x 12 – 35 = 25
T = 1 + 2 + 3 + 8 + 14 = 25
I have found the formula for the 5x5 grid. I am now going to find the formula for the 4x4 grid.
4x4 grid
T total = T + T-4 + T-9 + T-8 + T-7
= 5T – 28
CHECK:
T total = 5 x 10 – 28 = 22
T = 1 + 2 + 3 + 6 + 10 = 22
Finally, I am going to find the formula for the 3x3 grid. I have noticed so far that when u decrease the grid size by 1, the number u decrease from the 5T, increases by 7. i.e. 5T – 28 <- for the 4x4 grid. Therefore I predict that the next formula will be 5T – 21.
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.
T stands for the T number
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