Patterns we can notice:
We can see clearly that there is a pattern and a relationship within these numbers. For every one the Tn goes up, the Tt goes up by five.
Using Algebraic Numbers to Find a Formula
In the Horizontal
I will express these T-Shapes in algebraic form, using the nth term, and see if I can find a pattern that applies to all of the T-shapes.
I will express the T-Number as ‘n’.
We can see a pattern that applies to both T-Shapes, found by using the nth term, and so we are going to use these T-shapes to help us find the formula. Remember that these are the T-shapes going horizontally.
Tt = n + (n-9) + (n-18) + (n-17) + (n-19)
If we multiply out the brackets, we get:
Tt = 5n – (9+18+17+19)
Or
Tt = 5n – 63
Remember: ‘n’ is the T-Number.
Testing Our Formula
Okay, so we have found a reasonable formula, so let’s test it, using the first T-shape in the 9x9 grid as an example:
Our formula: Tt = 5n – 63
In this case, n = 20, and the Tt = T-Total.
5 x 20 = 100
- 63 = 37
Now let’s find the T-total and see if our formula works properly.
T-total (Tt) = 1 + 2 + 3 + 11 + 20
= 37.
It works!!!!
Just to make sure that there are no flaws, we will test this formula again with another shape on the horizontal, and then go on to the vertical, and see if a new formula is needed, or there is some kind of relationship in the formulae.
Our formula: Tt = 5n – 63
In this case, n = 21.
5 x 21 = 105
- 63 = 42
Now let us find the T-total and see if our formula works properly.
T-total (Tt) = 2 + 3 + 4 + 12 + 21
= 42
It does work!
Okay so we have established a formula that works for T-Shapes that have been translated horizontally in a 9x9 grid as far as we know. And we will now go on and try this with verticals, starting again with the process of finding as many T-Shapes that we can, overlapping as many times as possible, recording our data into a table, analyzing our findings, and finding any relationships or patterns within these figures.
Verticals
1st T-Shape
1st T-Shape:
T-Total
1 + 2 + 3 + 11 + 20
= 37
T-Number
= 20
2nd T-Shape
2nd T-Shape:
T-Total
10 + 11 + 12 + 20 + 29
= 82
T-Number
= 29
3rd T-Shape
3rd T-Shape: T-Total
19 + 20 + 21 + 29 + 38
= 127
T-Number
= 38
4th T-Shape
4th T-Shape:
T-Total
28 + 29 + 30 + 38 + 47
= 172
T-Number
= 47
5th T-Shape
5th T-Shape:
T-Total
37 + 38 + 39 + 47 + 56
= 217
T-Number
= 56
6th T-Shape
6th T-Shape:
T-Total
46 + 47 + 48 + 56 + 65
= 262
T-Number
= 65
7th T-Shape
7th T-Shape:
T-Total
55 + 56 + 57 + 65 + 74
= 307
T-Number
= 74
Recordings & Findings
I have collected data from all of the possible T-shapes vertically. I will now record my findings into a table, and see what patterns I can find, and then determine whether there is a link between the T-number and the T-total.
Patterns we can notice:
We can see clearly that there is a pattern and a relationship within these numbers. For every nine the Tn goes up, the Tt goes up by forty-five.
We can understand that the numbers go up nine times as much vertically, than they do horizontally, because there are 9 rows in the grid, and the numbers increase horizontally by one. By moving the T-Shape on the vertical, each number increases by 9 each time, because it is a 9x9 grid.
This means that in the horizontal if the T-number went up by 1 each time, in the vertical T-Shapes, the T-Number would go up by 1 x 9 = 9 each time, and we apply the same thing for the T-Total; 5 x 9 = 45 each time.
This works out correct.
Using Algebraic Numbers to Find a Formula
In the Vertical
I will express these T-Shapes in algebraic form, using the nth term, and see if I can find a pattern that applies to all of the T-shapes.
We can see a similar pattern that applies to both T-Shapes, using the nth term, and so we are going to use these T-shapes to help us find the formula. Remember that these are the T-shapes going vertically.
Tt = n + (n-9) + (n-18) + (n-17) + (n-19)
If we multiply out the brackets, we get:
Tt = 5n – (9+18+17+19)
Or
Tt = 5n – 63
Remember: ‘n’ is the T-Number in this equation.
We can see straight away that the same formula can be applied for both the horizontal and the vertical, but just to check, I will try to use this formula on a T-Shape in a random position on the 9x9 grid:
T-Total
33 + 34 + 35 + 43 + 52
= 197
T-Number
= 52
Now that we know the T-Total and T-Number, we can test our formula and see what our results may come to be.
Using this formula: Tt = 5 x Tn – 63,
The T-Total should work out as:
5 x 52 – 63
= 260 – 63
= 197
We have now established that on a 9x9 grid, any T-Total can be found using the formula:
Tt = 5n – 63
Where n is the T-Number.
Ok so that’s sorted, now let us see if we can find a formula using the relationship between the T-number (t) and the grid size* (g).
g = 9, the width of the grid we are looking at.
As we can see from the above T-Shapes, when trying to find a relationship with the grid size 9x9, using the nth term, we come up with a long formula, which we are about to break down.
T-total = n + (n – g) + (n – 2g) + (n – 2g – 1) + (n – 2g + 1)
Or Tt = 5n – 7g
This generalised formula is very central to our investigation, as it should now be able to work on grids of other sizes, and I will demonstrate it in action on this 4x4 grid.
So in this test, (g) would become 4, as we are using a 4x4 grid.
Tt = (5 x 15) – 7(4)
= 75 – 28
= 47
T-Total = 47
We have a correct answer, but just to check it is not a one off, we will repeat check this formula again in an 8x8 grid as follows:
Tt = (5 x 47) – 7(8)
= 235 – 56
= 179
T-total = 30 + 31 + 32 + 39 + 47 = 179
Any T-Total of a T-Shape can be found if you have the 2 variables of a T-Number and a grid size for T shapes that extends upwards, using the formula Tt = 5n – 7g.
Finding relationships on different sized grids
T-Total
19 + 20 + 21 + 28 + 36
= 124
T-Number
= 36
If we take this 8x8 grid, and try to generalize this straight away, using the same methods that were used before for the 9x9 grid, we achieve the formula:
Tt = n + n – 8 + n – 17 + n – 16 + n – 15
Tt = 2n – 8 + 3n – 48
Tt = 5n – 56
This can also be shown using the nth term, yet again:
Testing this formula, using n as 36, we get:
Tt = (5 x 36) - 56
= 180 – 56
Tt = 124
This gives us yet again the right answer, which proves that this formula is working.
Another pattern i have found is that, the number we subtract from the T-Total (56 in the above case) is 7 x (g), g being the grid size. I also noticed this pattern in the 9x9 grid when we took away 63 within our formula, and 7 x 9 = 63 (9 was our grid size).
Now I will repeat this on a 4x4 grid, and compare my findings in a table.
On this 4x4 grid, we can use the same generalised formula to find out the T-Total:
Tt = n + n – 4 + n – 9 + n – 8 + n – 7
Tt = 2n – 4 + 3n – 24
Tt = 5n – 28
If we test this:
Tt = (5 × 15) – 28
Tt = 75 – 28
Tt = 47
We get the right answer, which shows how accurate this formula is, and that it is reliable when it comes to any grid sizes.
Here is a table with a pattern that I mentioned earlier, which I had noticed went up in multiples of seven.
Therefore, it is only fair to assume that the rest of this table would look like this:
Now we have established a firm pattern that we can use throughout any regular even-sized grid, looking at this table, we can find the formula to any one of these grids.
For example:
11x11 ➔ Tt = 5n - 77
10x10 ➔ Tt = 5n - 70
7x7 ➔ Tt = 5n - 49
6x6 ➔ Tt = 5n - 42
All formulas for finding the T-Total (as long as the grid width and height are equal) are 5 x (T-Number) – 7 x Grid Width (g)
We can see this working for the above like so:
11x11 ➔ Tt = 5n – 77 (11 x 7)
10x10 ➔ Tt = 5n – 70 (10 x 7)
7x7 ➔ Tt = 5n – 49 (7 x 7)
6x6 ➔ Tt = 5n – 42 (6 x 7)
These are all the right formulae.
Rotations
We have only tried to find formulae for T-Shapes that can be moved either across, down, or both. Now we are going to find out what kind of changes (if any) take place within the formulas, if the T-Shape we are investigating, is rotated about a certain point.
Again I am going to go back to my original 9x9 grid, and use this to show how a T-Shape can be rotated, 90°, 180°, and 270° both clockwise and anticlockwise, around the T-Number, as we can see here; Firstly; 90° clockwise:
The table above shows our T-Shape being rotated 90° clockwise
T-Number of rotated T-Shape: 30
T-Total of rotated T-Shape: 30 + 31 + 23 + 32 + 41 = 157
I will now replace the numbers within the rotated T-Shape, with algebraic terms (n’th term), and then using the table width/grid size (g), I will try to find a generalised formula, so here we go:
If we simplify this equation, we can find the general formula that might apply to any T-Shape rotated on its side 90 degrees clockwise.
Tt = n + (n + 1)+ (n – (g - 2)) + (n + 2) + (n + (g + 2))
General Formula ➔ Tt = 5n + 7
We already know that the T-Number is 30 and the T-Total is 157, but let us try to work that out using our newly found formula:
Tt = (5 x 30) + 7
= 150 + 7
= 157
This is the correct answer. This shows that this formula works, but just to make sure that we can use this formula with any other sized grid; I am going to test it again, this time using an 8x8 grid.
The table above shows our T-Shape being rotated 90° clockwise
T-Number of rotated T-Shape: 36
T-Total of rotated T-Shape: 36 + 37 + 38 + 30 + 46 = 187
So let us try the general formula we have just discovered:
Tt =5 x 36 + 7
= 180 + 7
Tt = 187
180°
The table above shows our T-Shape being rotated 180°, about 30 as the centre of rotation on a 9x9 grid.
T-Number of rotated T-Shape: 30
T-Total of rotated T-Shape: 30 + 39 + 48 + 47 + 49 = 213
I will now replace the numbers within the rotated T-Shape, with algebraic terms (nth term), and then using table width/grid size (g) to try to find a general formula, so here we go:
If we simplify this equation, we can find the general formula that might apply to any T-Shape rotated 180 degrees clockwise.
Tt = n + (n + g) + (n + 2g) + (n + (2g + 1)) + (n + (2g – 1)
General Formula ➔ Tt = 5n + 7g
We already know that the T-Number is 30 and the T-Total is 213, but let us try to work that out using our newly found formula:
Tt = (5 x 30) + 7(9)
= 150 + 63
= 213
This is the correct answer. This shows that this formula works, but just to make sure that we can use this formula with any other sized grid; I am going to test it again, this time using an 8x8 grid.
The table above shows our T-Shape being rotated 180°.
T-Number of rotated T-Shape: 36
T-Total of rotated T-Shape: 36 + 44 + 51 + 52 + 53 = 236
So let us try the general formula we have just discovered:
Tt =5 x 36 + 7(8)
= 180 + 56
Tt = 256
270° Clockwise
The table above shows our T-Shape being rotated 270° clockwise, about 30 as the centre of rotation on a 9x9 grid.
T-Number of rotated T-Shape: 30
T-Total of rotated T-Shape: 30 + 29 + 28 + 37 + 19 = 143
I will now replace the numbers within the rotated T-Shape, with algebraic terms (nth term), and then using table width/grid size (g) to try to find a general formula, so here we go:
If we simplify this equation, we can find the general formula that might apply to any T-Shape rotated 270 degrees clockwise.
Tt = n + (n - 1) + (n - 2) + (n + (g – 2)) + (n - (g + 2))
General Formula ➔ Tt = 5n - 7
We already know that the T-Number is 30 and the T-Total is 143, but let us try to work that out using our newly found formula:
Tt = (5 x 30) - 7
= 150 - 7
= 143
This is the correct answer. This shows that this formula works, but just to make sure that we can use this formula with any other sized grid; I am going to test it again, this time using an 8x8 grid.
The table above shows our T-Shape being rotated 270° clockwise.
T-Number of rotated T-Shape: 36
T-Total of rotated T-Shape: 36 + 35 + 34 + 42 + 26 = 173
So let us try the general formula we have just discovered:
Tt =5 x 36 - 7
= 180 - 7
Tt = 256
Constraints
Throughout this investigation, I have come across many different sized grids and have worked with differently transformed T-Shapes within these grids. I have found that it is not always possible to draw a T-Shape anywhere within a grid, and I will show you some examples:
In a 3x3 grid, we can draw this T-Shape:
But not this one:
If we expand this to a bigger table, as follows, we can notice many scenarios when drawing a T-Shape would be impossible, and I shall show and list them. The shapes with dotted outlines do not work.
As we can see, the T-Shapes that have been dotted, do not work, simply because they cannot fit all the squares they require to make a T, into this 9x9 grid.
Now let me put that into mathematical language:
- We cannot draw a complete T-Shape if the T-Number of that T-Shape is either:
- Contained in the first two rows of a grid, be it any size.
- Contained in the first or the last column of a grid, be it any size.
Therefore, after coming to this conclusion, we can see that it would be impossible for any of our formulae to work, if our T-Number is situated on one of the above anomalies.
We conclude that to find the formula to any T-Shape in a grid, the T-Number of that shape must not be: a) contained in the first two rows of that grid, or b) contained in the first or last column of that grid.
Conclusion
To conclude, throughout this investigation I have analyzed T-Shapes in different grid sizes and systematically (step-by-step), using many different symbols and Geometrical language relating to the T-Shape problem, to help me find a generalised formula, for working out a relationship between T-Numbers and T-totals, taking into account grid sizes, transformations, and rotations.
Throughout the investigation, which I have now conducted, I became more and more aware of the reliability of my findings, by testing my predictions, using trial and error, and checking and re-checking, until I could confirm that a reliable pattern could be established.
At the end of the day, I discovered that no matter what, after you find one generalised formula, you can find the formula to a lot of things that are along the same lines, very easily.
This investigation was useful to me because it helped me develop my ability to find formulas using algebra, and generalise them, which personally I thought was one of my weaknesses in Maths, but it was a skill I had to draw upon during this investigation, as it was very central to what we were looking at. It has very much improved my ability. Also, this coursework has helped my mathematical vocabulary and the use of my mathematical language overall.
Thank You.
