What the 'L' - L shape investigation.
Alexander Ford 11L
WHAT THE 'L'
Mathematics GCSE
Coursework
L Shape Investigation
The diagram below shows a standard L shape, which is drawn on a 9 by 9 number grid.
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81
The total of the numbers inside the L-Shape is 1 + 10 + 19 + 20 + 21 = 71.
This is called the L-Sum.
The number in the bottom left hand corner of the L-Shape is called the L-Number.
The L-Number for this L-Shape is 19.
This investigation is in three parts.
PART 1 Is to investigate the relationship between the L-Sum and the L-Number.
PART 2. Using grids of different sizes is to translate the L-Shape to different positions. Investigate the relationship between the L-Sum, the L-Numbers and the grid size.
PART 3. Investigate what happens if you use L-Shapes of different sizes. Try other transformations and combinations of transformations. Investigate relationships between the L-Sum, the L-Number, the grid size and the transformations.
I will use the following key to illustrate the algebraic terms that I will use during my investigation:
KEY
L
L-Number
C
Number of cells in L-Shape
G
Grid Size
?
The Sum Of
X
Number Of Cells Above The L-Number
Y
Number Of Cells Across From The L-Number
To start my investigation I will use the 9 by 9 grid section above I calculated the sum of the first seven L-Shapes; shown below:
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9
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I calculated up to seven three by three L-Shapes and found their L-Sum (as this is the last L-Shape that will fit across a nine by nine grid). Numbers eight and nine and all other numbers in those columns will not be able to produce an L-Shape as the across axis arm will not allow an entire L-Shape. The next L-Shape number in the grid would be 28.
I will display my results from these calculations into a table format as follows:
Number In Sequence
2
3
4
5
6
7
L-Sum
71
76
81
86
91
96
01
I will now use the difference method to establish and determine if a sequence pattern exists between them. Therefore, the next step is to calculate the differences between each L-Shape.
Number In Sequence
2
3
4
5
6
7
L-Sum
71
76
81
86
91
96
01
Difference
5 5 5 5 5 5
As the first part of my investigation is to find the relationship between the L-Sum and the L-Number I am going to times the difference of the L-Sum with the L-Number. Thus giving me 5L.The results show that the 1st difference is constant and therefore, the formula must consist of 5L.
My results shown in table format are as follows:
Number in sequence
5L
Difference
Sum of L-Shape
(L-Sum)
95
-24
71
2
00
-24
76
3
05
-24
81
4
10
-24
86
5
15
-24
91
6
20
-24
96
7
25
-24
01
From the table of results I can see that the difference between 5L and the L-Sum is a constant -24. Therefore, we have 5L -24 present in the formula. I will use this formula to prove its correctness and to look further for additional differences.
Number In Sequence
Formula
Formula Equation
Results
L-Sum
5L -24
(5 x 19) - 24
71
71
2
5L -24
(5 x 20) - 24
76
76
3
5L -24
(5 x 21) - 24
81
81
4
5L -24
(5 x 22) - 24
86
86
5
5L -24
(5 x 23) - 24
91
91
6
5L -24
(5 x 24) - 24
96
96
7
5L -24
(5 x 25) - 24
01
01
From these answers we can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 9 by 9 grid. Therefore, the algebraic formula to ...
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(5 x 21) - 24
81
81
4
5L -24
(5 x 22) - 24
86
86
5
5L -24
(5 x 23) - 24
91
91
6
5L -24
(5 x 24) - 24
96
96
7
5L -24
(5 x 25) - 24
01
01
From these answers we can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 9 by 9 grid. Therefore, the algebraic formula to calculate an L-Sum given the L-Number in a standard L-Shape in a 9 by 9 grid is:
5L -24
To prove my formula I am going to utilise the L-Shape, replacing the numbers with letters to prove the formula using any number in a 9 by 9 grid.
L-18
L-9
L
L+1
L+2
*NB: Note that in this L-Shape numbers along the horizontal will increase and numbers on the vertical will decrease.
Also note that the number vertically up from the L-Number is referred to as L-9, the second is L-18.
Note that 9 is the grid square and 18 is double the grid size and two rows up.
The sum of the shape above gives me the last part of the formula, which is -24:
+ 2 + (-9) + (-18) = -24
Therefore, the formula of 5L - 24 is correct.
I have proved that my formula works and using my results I can accurately predict any L-Sum given any L-Number in a 9 by 9 grid.
PART 2
I will now use grids of different sizes, translate the L-Shape to different positions and investigate the relationship between the L-Sum, the L-Numbers and the grid size.
Using the difference method as used in Part 1 on 4 by 4 grid I have found:
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9
0
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6
I will display my results from these calculations into a table format as follows:
Number In Sequence
2
3
4
L-Sum
36
41
56
61
I will now use the difference method to establish and determine if a sequence pattern exists between them. Therefore, the next step is to calculate the differences between each L-Shape.
Number In Sequence
2
3
4
L-Sum
36
41
56
61
Difference
5 * 5
* The difference between Number In Sequence 2 & 3 is not 5 as the L-Shapes form in different rows.
As part of my investigation is to find the relationship between the L-Sum and the L-Number I am going to times the difference of the L-Sum with the L-Number. Thus giving me 5L.The results show that the 1st difference is constant and therefore, the formula must consist of 5L.
My results shown in table format are as follows:
Number in sequence
5L
Difference
Sum of L-Shape
(L-Sum)
45
-9
36
2
50
-9
41
3
65
-9
56
4
70
-9
61
From the table of results I can see that the difference between 5L and the L-Sum is a constant -9. Therefore, we have 5L -9 present in the formula. I will use this formula to verify its correctness and to look further for additional differences.
Number In Sequence
Formula
Formula Equation
Results
L-Sum
5L -9
(5 x 9) - 9
36
36
2
5L -9
(5 x 10) - 9
41
41
3
5L -9
(5 x 13) - 9
56
56
4
5L -9
(5 x 14) - 9
61
61
From these answers I can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 4 by 4 grid. Therefore, the algebraic formula to calculate an L-Sum given the L-Number in a standard L-Shape in a 4 by 4 grid is:
5L -9
To prove my formula I am going to utilise the L-Shape, replacing the numbers with letters to prove the formula using any number in a 4 by 4 grid.
L-8
L-4
L
L+1
L+2
*NB: Note that in this L-Shape numbers along the horizontal will increase and numbers on the vertical will decrease.
Also note that the number vertically up from the L-Number is referred to as L-4, the second is L-8.
Note that 4 is the grid square and 8 is double the grid size and two rows up.
The sum of the shape above gives me the last part of the formula, which is -9:
+ 2 + (-4) + (-8) = -9
Therefore, the formula of 5L - 9 is correct.
I have proved that my formula works and using my results I can accurately predict any L-Sum given any L-Number in a 4 by 4 grid.
Using the difference method as used in Part 1 on 5 by 5 grid I have found:
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I will display my results from these calculations into a table format as follows:
Number In Sequence
2
3
4
5
6
L-Sum
43
48
53
68
73
78
I will now use the difference method to establish and determine if a sequence pattern exists between them. Therefore, the next step is to calculate the differences between each L-Shape.
Number In Sequence
2
3
4
5
6
L-Sum
43
48
53
68
73
78
Difference
5 5 * 5 5
* The difference between Number In Sequence 3 & 4 is not 5 as the L-Shapes form in different rows.
As part of my investigation is to find the relationship between the L-Sum and the L-Number I am going to times the difference of the L-Sum with the L-Number. Thus giving me 5L.The results show that the 1st difference is constant and therefore, the formula must consist of 5L.
My results shown in table format are as follows:
Number in sequence
5L
Difference
Sum of L-Shape
(L-Sum)
55
-12
43
2
60
-12
48
3
65
-12
53
4
80
-12
68
5
85
-12
73
6
90
-12
78
From the table of results I can see that the difference between 5L and the L-Sum is a constant -12. Therefore, we have 5L -12 present in the formula. I will use this formula to verify its correctness and to look further for additional differences.
Number In Sequence
Formula
Formula Equation
Results
L-Sum
5L - 12
( 5 x 11) - 12
43
43
2
5L - 12
( 5 x 12) - 12
48
48
3
5L - 12
( 5 x 13) - 12
53
53
4
5L - 12
( 5 x 16) - 12
68
68
5
5L - 12
( 5 x 17) - 12
73
73
6
5L - 12
( 5 x 18) - 12
78
78
From these answers we can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 5 by 5 grid. Therefore, the algebraic formula to calculate an L-Sum given the L-Number in a standard L-Shape in a 5 by 5 grid is:
5L -12
To prove my formula I am going to utilise the L-Shape, replacing the numbers with letters to prove the formula using any number in a 5 by 5 grid.
L-10
L-5
L
L+1
L+2
*NB: Note that in this L-Shape numbers along the horizontal will increase and numbers on the vertical will decrease.
Also note that the number vertically up from the L-Number is referred to as L-5, the second is L-10.
Note that 5 is the grid square and 10 is double the grid size and two rows up.
The sum of the shape above gives me the last part of the formula, which is -12:
+ 2 + (-5) + (-10) = -12
Therefore the formula of 5L - 12 is correct.
I have proved that my formula works and using my results I can accurately predict any L-Sum given any L-Number in a 5 by 5 grid. I have also proved that 5L is common to all of my calculations using a standard five cell L-Shape. The algebraic formula calculated from the L-Shape has allowed me to calculate the final parts of my formulae given the grid size. Using this information I can predict the formulas for a 6 by 6, 7 by 7 and 8 by 8 grid.
Grid Size
Algebraic L-Shape
Formula
6 by 6
L-12
L-6
L
L+1
L+2
5L - 15
7 by 7
L-14
L-7
L
L+1
L+2
5L - 18
8 by 8
L-16
L-8
L
L+1
L+2
5L - 21
I am now going to find a formula to find the L-Sum given just the L-Number and grid size. The formula must also be able to be used in any size grid.
By looking at all of the different grid sizes and their formulae. I can note that they all start with 5L; therefore, my final formula must consist of 5L.
Grid Size
Final Part Of The Formula
Difference
4 by 4
-9
5 by 5
-12
6 by 6
-15
7 by 7
-18
8 by 8
-21
9 by 9
-24
By looking at the results in my table I can see that the difference is always 3 and as Part 2 of this investigation is to find a relationship between the L-Sum, the L-Shape and the grid size I am going to times the three by the grid size (g) and using the difference method to see if there is any relationship between the L-Sum, L-Shape and grid size.
3g
Difference
Last Part Of The Formula
(3 x 4) 12
-3
9
(3 x 5) 15
-3
2
(3 x 6) 18
-3
5
(3 x 7) 21
-3
8
(3 x 8) 24
-3
21
(3 x 9) 27
-3
24
Looking at the table above I have a constant of -3. This suggests that -3 is to be added to the formula. From these two tables so far I have the formula 3g - 3. I will use this formula to prove its correctness and to look further for additional differences.
3g - 3
Answer
Last Part Of The Formula
(3 x 4) - 3
9
9
(3 x 5) - 3
2
2
(3 x 6) - 3
5
5
(3 x 7) - 3
8
8
(3 x 8) - 3
21
21
(3 x 9) - 3
24
24
Thus my formula so far is as follows: -
( 5L ) + (-3g + 3)
The two parts of the formula are in brackets, as each part has to be calculated separately and then added together. By looking at the results in my table I have noticed that each formula is 5L minus a value. Using the plus and minus rule I have applied this rule to the formula. Therefore, -3g + 3 is equal to 3g - 3.
I will now prove that my formula works. If I use the same algebraic L-Shape as before I can prove my formula works.
L-2g
L-1g
L
L+1
L+2
To enhance my calculations I have replaced the numbers above the L-Number with algebraic notation. I said before that the number one row up from the L-Number was the grid size and that the number two rows up was double the grid size. Now the L-Shape works in any size grid. As before I stated that the sum of the axis was equal to the last part of the formula. This is true for this L-Shape and I will now calculate the sum of the axis to prove it.
(-1g) + (-2g) + (1) + (2) = -3 + 3
Thus, I have proved that the formula is correct.
Therefore the formula for a standard 5 cell L-Shape in different positions in different sized grids is: -
( 5L ) + (-3g + 3)
PART 3
I will now use grids and L-Shapes of different sizes; I will try more transformations and combinations of transformations. I will look into the relationship between the L-Sum, the L-Number, the grid size and the transformations.
In the previous formulae I was using a standard 5 cell L-Shape and the formulae consisted of 5L. Therefore, if the amount of cells in the L-Shape is to change then it will become a variable instead of a constant. Consequently the number 5 will have to be replaced with a letter to allow it to become a variable, for the sake of argument I will now call this part of the equation (CL).
In the previous part of this investigation I mentioned that the sum of the X-axis added to the sum of the Y-axis gave me the final part of the formula. Hence for ease of notation I will call the sum of the X-axis ?x and the sum of the Y-axis ?y. Therefore, ?x + ?y is equal to the final part of the formula. As in part 2 of my investigation each part to the formula has to be worked-out separately and then added together. Therefore, at the moment my formula will consist of: -
(CL) + ( - ( ?x ) + ( ?y ))
This formula would only allow me to calculate the L-Sum using any sized L-Shape. However, it wouldn't allow me to change the grid size. Therefore, to allow my formula to work in any sized grid I have to change the grid size into a variable and to do this I am going to call it g. In Part 2 of this investigation I used a pure algebraic L-Shape to calculate the final part of the formula. By examining the shape more carefully I have found that the up-axis is always multiplied by g. This tells me that my up-axis in the formula will have to be multiplied by g to allow it to work in any sized grid. So far my formula consists of: -
(CL) + (- ( ?x )g + (?y))
I will use the same symbols that I used in the formula in Part 2 and will be applying the plus and minus rule.
E.g.
( 5L ) + ( -3g + 3 ) is equal to (CL) + (- ( ?x )g + (?y))
Researching the new symbol ? told me that I needed to use ranges on this symbol to make it mathematically correct. The range is a set of numbers above and below the figure that tell one where the sequence starts and were it finishes.
Therefore, so far in my investigation I have: -
Through further research I have found that the "sum of" sign can be replaced with: -
This is known as Gausses' Theory.
Using this new knowledge my formula has changed to: -
After testing this formula my results were as follows: -
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Letter
Value
Equation
Answer
C
7
365
CORRECT
L
59
G
9
X
3
Y
3
My formula now works on any size grid with any size L-Shape with any size arms at 0° rotation angle.
I will now try and extend my investigation to make my formula so that it works in any size grid with any size L-Shape with any size arms and now in any rotation.
To do this I am going to look at the change in the L-Shape when it is rotated. I will start by doing a diagram of the four rotations that are possible using a standard L-Shape.
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These are the four rotations you would get if you rotated a standard L-Shape in a 9 by 9 grid. The four L-Shapes below are my results: -
23
32
41
42
43
Rotating the L-Shape through 90°
41
42
43
50
59
23
32
39
40
41
39
40
41
50
59
Rotating the L-Shape through 180°
Rotating the L-Shape through 270°
I will now tabulate my results to make it easier to read.
Arm
0°
90°
80°
270°
X
-
+
+
-
Y
+
+
-
-
With the above information I can predict that if the rotation of the L-Shape is changed then the + & - in the equation will have to change. Therefore, the equation to find the L-Sum given just the grid size and the L-Number, in any size grid, in any rotation, with any transition of the L-Shape, with any size arms is: -
Now, when using this formula you have to choose a rotation look at the signs needed for that rotation and replace the ± signs with the correct symbol.
Letter
Value
Rotation Angle
+ & -
Equation
Answer
C
7
0°
X= -
Y= +
365
CORRECT
L
59
G
9
X
3
Y
3
Letter
Value
Rotation Angle
+ & -
Equation
Answer
C
7
80°
X= +
Y= -
461
CORRECT
L
59
G
9
X
3
Y
3
Thus, I have proved that my formula works for any rotation.
For an even further extension I have created an excel spreadsheet that allows my to calculate the L-Sum given just C, L, G, X, Y and the rotation angle using the above formula. A screen dump below shows what the package looks-like and what formulas I have used. The spread sheet is also included on the disc attached.