E.g. T number +1 = T total +5 this would only work for moving the T number on space to the right I need a formula that can calculate T total from any given T number.
I can now try to make an algebraic expression for T Number inside a 9x9 grid.
Make N = T Number
The rule I found is:
T-Total = (N-19) + (N-18) + (N-17) + (N-9) + (N)
I then can simplify this to:
T-Total = 5n-63
I will now test my expressions to see if it works correctly
For example if n=40:
T-Total= (5x40)-63
T-Total=200-63
T-Total=137
The nth term = 5N – 63
To find the nth tern I looked at the T – Number value and times it by the gap which is 5 and then subtracted by the first T-number to get the gap which in this case was 63.
The formula can be used on any of the T number in the 9x9 grid. I have found a general rule which will work out the T total of a given T number. Except for the numbers that this rule cannot be used because the T shape does not fit.
9x9-14x14
Now I have worked out a formula for calculating T total from T number on a 9x 9 grid, I will now try to do this on other grid sizes using the same method as before, ranging from 9x9-14x14 and record my results in a table.
T-Total = (N-19) + (N-18) + (N-17) + (N-9) + (N)
I then can simplify this to:
T-Total = 5n-63
T-total = (N-21)+ (N-20) + (N-19) + (N-10)+( N)
I then can simplify this to:
T-total = 5N-70
T-total = (N-23)+(N-22)+(N-21)+(N-11)+(N)
I then can simplify this to:
T-total = 5N-77
T-total = (N-25)+(N-24)+(N-23)+(N-12)+(N)
I then can simplify this to:
T-total = 5N-84
T-total = (N-27)+(N-26)+(N-25)+(N-13)+(N)
I then can simplify this to:
T-total = 5N-91
T-total = (N-29)+(N-28)+(N-27)+(N-14)+(N)
I then can simplify this to:
T-total = 5N-98
Using the results from the other grid sizes I have constructed the table below
Generalisation
Now I have found rules for grid sizes 9-14 I will now try and find a rule which can find the t total of any grid size.
I noticed that every time the grid size is increased by 1 the variable at the end of the formula becomes another minus seven to what it already was before, e.g. 5N-21, 5N-28, 5N-35…… and so on. The constant value is always 5N, simply because there are always 5 numbers in the T-Shape. I also noticed a pattern between the grid size and the value to be subtracted at the end of the equation.
From the table I can work out a simple formula to find the T-Total of any sized grid. Using another T-Shape will help me prove the formula:
(Using a 3 by
3 grid as an
Example)
Where G is the grid size (in this example G = 3)
When worked out:
T-Total = (N-2G-1) + (N-2G) + (N-2G+1) + (N-G) + (N
= 5N-7G
Therefore the equation for finding the T-Total for any grid size is:
Using the formula above I can work out the following: (although this is not possible as a 3 by 3 sized grid is necessary for a T-Shape) the equation would be 5N-7, for a 2 by 2 (again not possible to do) the equation would be 5N-14, for a 3 by 3 it would be 5N-21, and so forth up to huge numbers that would be very hard to work out using arithmetic – for example 231 by 231 would have a T-Total equation of 5N - (231x7) which works out as 5N-1617. Similarly a 45 by 45 grid size would work out as 5N- 315 and a 74 by 74 grid size would have an equation of 5N-518. now I will move onto rotations.
Rotations
When rotating the T shape 90° to the right. It is the exact same process as working out the formula for the first 9x9 grid I did first. The only difference is that the T is on its side.
I began this part by rethinking my initial formula. If I am to test the T-Shape in different rotations of 0˚, 90˚, 180˚ and 270˚ then the formula ‘5N-7G’ will only work for 0˚ as this is the angle the formula was based upon.
I next drew the T-Shape rotated 90˚ in the right on a 6x6 grid as shown below
The T-Number of this shape is 14 and the T-Total is 77.
I have used the same method below as I have in the other grids.
=
T-Total for 90˚ rotations on a 6x6 grid = (N) + (N+1) + (N-4) + (N+2) + (N+8)
This can be simplified to:
= 5N + 7
I then tested this formula on another shape in the 6 by 6 table, using the numbers 8, 9, 10, 4 and 16.
The calculated T-Total for this shape is 32, with the T-Number being 8. When putting 8 into the formula I’d just calculated I got 5(8) + 7 which equals 32 also.
Before moving onto the next step I tested the formula on grids of different sizes. I chose a 4 by 4 grid to test the formula on:
The calculated T-Total for this shape is 32, To prove this I checked it into the formula and came out with 5(5) + 7 which equals 32. I can therefore prove that:
I then used simple logic to work out the formula for a 270˚ rotation. If 90˚ was 5N + 7 and the angle is 90˚ in the other direction (270˚ of a complete 360˚ rotation) then the formula must be 5N-7. I tested and tried this formula again on a 6 by 6 grid:
The T-Total for this shape is 42 when calculated. When put into the formula it becomes 5(7) + 7 which also equals 42. This proves my theory that:
As with the last formula, I used simple logic to solve the 180˚ rotation formula. If the formula for 0˚ is 5N -7G then the logical formula to solve the 180˚ rotation would be 5N + 7G. I tried and tested the theory below:
The calculated T-Total for this T-Shape is 57 and the T-Total when put into the formula above also equals 57. This proves that
Conclusion
In this investigation I have successfully established a relationship between T-Total and Grid size, and found a rule for grids ranging from 9-14. I have also found a general rule which will find the T-total for any grid size. I then move onto rotations I looked at 90˚, 180˚, 270˚ and 360˚, I worked out my formulas for rotating the T-shape on a 6x6 grid and successfully found and tested and proved my formula correct. Overall I think the investigation went well.
Evaluation
If I was to do the investigation again I would do grid rotations on larger grid sizes and make a general rule for rotating the T- shape.
