T-total = 7+8+9+13+18 = 55
Using the formula: 5x18=90 – (7x5) = 55
The formula works for this grid size as well.
Translations: across
I will now investigate the formula for T-numbers when it is translated.
When the T-shape is moved across 1, all the numbers go up by 1.
The same rule applies for when it is moved across1, the numbers also go up by 2.
Formula:
N+A
Variable = A
A= the number of times T is moved across
Translations: down
When you move the T-shape down one square the number increases by the size of the grid. For example, in this grid, every number increases by 9.
When the T-shape is moved 2 squares down, the number increases by 2x size of the grid.
The same rule applies: When T-shape is moved 3 squares down, the number increases by 3x size of the grid.
Formula:
N= T-number
D= the amount of times the T-shape is moved down.
G= Grid size
The formula which works for any grid size is: N+DG
Translation: across and down
Formula for T-number: N+A+DG
Formula for T-total:
T-total = N+A+DG-2G-1+N+A+DG-2G+N+A+DG-2G+1+N+A+DG-G+N+A+DG
= 5N+5A+5DG-7G
T-total factorised = 5(N+A+DG)-7G
Checking the formula:
Limitations:
5x5 grid: We are able to move the shape across 2 squares maximum.
6x6 grid: We are able to move the shape across 3 squares maximum.
7x7 grid: We are able to move the shape across 4 squares maximum.
I have noticed a pattern emerging. In all three grids that are shown above, the limitation of the T-shape is the grid size minus 3. For example, in the 5x5 grid: 5 is the grid size, 2 is the limitation. 5-2 = 3
In the 6x6 grid: 6 is the grid size, 3 is the limitation, 6-3 = 3
Therefore, the formula for limitations of the T-shape is g-3.
Rotation:
N’ = new n number
T-total for a rotated shape = n’+n’+1+n’+2+n’+2+g+n’+2-g
T-total = 5n’+7
Formula for this grid: n’ = n+1+2g+2-g
N’ = 18+1+16+2-8
N’ = 29
General formula for any grid:
N’ = n + h + vg + v – hg
Translation and Rotation:
The formula for translation and rotation is just the formula for translation and the formula for rotation together.
Therefore it would be N+A+DG+H+VG+V-HG = New t-number
7x7 grid: translation: 1 across, 3 down. Centre point: 1 horizontal and 1 vertical away from original n number.
My prediction: 16+1+ (3x7) = 38
38+1+7+1-7 = 40
40= New t-number
Checking my prediction:
Red shape – original shape
Purple shape – translated shape
Orange shape – rotated shape
N’= new n number
In the grid shown above you can see that my prediction is correct. The new t-number after the shape has been translated and rotated is 40.
Therefore, N’ = N+A+DG+H+VG+V-HG.
The T-total for a shape which has been translated then rotated is as follows:
Therefore the T-total would be all the boxes added together.
T-total = 5N+5A+5DG+5H+5VG+5V-5HG+7
This can be factorised to get
T-total = 5(N+A+DG+H+VG+V-HG) +7
Now I will test this for the above grid to check if this works.
Formula:
5(N+A+DG+H+VG+V-HG) +7
Therefore: - 5(16+1+21+1+7+1-7) +7 = 207
T-total = 40+41+42+35+49 = 207
The formula works!
In conclusion to work out the T-number for a new shape after it has been translated and rotated you use the formula:
N’= N+A+DG+H+VGV-HG
And the formula for the T-total after the shape has been translated and rotated is:
N’= 5(N+A+GD+H+VG+V-HG)+7
