The object of this coursework is to find the relationship between the total value and the positioning of a T-shape on a number grid. I will investigate the following: What happens as the T-shape moves down the grid.

Introduction

The object of this coursework is to find the relationship between the total value and the positioning of a T-shape on a number grid. I will investigate the following:

What happens as the T-shape moves down the grid.
What happens as the T-shape moves across the grid.
What is the smallest possible value for the T-number.
What is the largest possible value for the T-number.
What happens when the T-shape is rotated 90°, 180° and 270°

Starting on a 10x10 grid, I will draw my T-shape and investigate the 5 tasks stated above. Then I will move on to do the same on a 9x9 and 8x8 grid to see if I can find a relationship. On each grid, once the T-shape is rotated, I will move the T across and down for each angle it’s rotated at, all the results shall be listed into tables. After this I will try to find the smallest and largest possible value for the T-shape. Finally I will compare my results and see if there are any patterns which will hopefully supply me with an algebraic formula that can be used on an ‘N’ sized grid.

Here is my 10x10 grid, I will draw my T-shape and rotate it 90°, 180° and 270°, each time it is rotated I will move the shape down and across the grid 3 times and record the results into a table.

T = N + N-10 + N-20 + N-21 + N-19

T = 5N = 70

This is a T-Shape on a 10 by 10 number grid. The total of the numbers inside the grid all add up to produce a T-Total. In this grid you add the numbers 1 + 2 + 3 + 12 + 13 which = 40. The N number is 22.

T = Total number in the shape

N = The number at the bottom

of the T (as shown)

Plan

1) I will start my investigation on a 10 by 10 grid. I will use the T-Shape in

the top left corner and work across.

2) I will then draw up a table of results and try and find a pattern in my

answers.

3) If I find a relationship in my answers I will then try and spot a rule for

the relationship between the T-Number and the T-Total on a 10 by 10

grid.

4) I will then test my rule.

5) I will then move the T-Shape down the grid. I will start in the top left

corner.

6) Next, I will move onto different grid sizes and repeat steps 1-5.

7) I will then try and find a general rule for the relationship between the T-

Number and the T-Total on any size grid.

8) I will then test my rule on a few grid sizes to check that my rule was not

just a coincidence.

9) Once I have made a rule, I will start to investigate the effect

transformations has on the T-Total.

10) I will finally write up a conclusion to sum up the investigation.

T-Totals

10 by 10 Grid

Question 1 (Down)

I am now going to begin my investigation on a 10 by 10 grid. I am going to answer question 1, what happens when the T-Shape moves across the grid?

The total value of this T-Shape is 40 (1 + 2 + 3 + 12 +22 = 40) and the N number is 22. To see if there is a sequence between the T-Shape, I will move the shape down one row.

The total value of this T-Shape is 90 (11 + 12 + 13 + 22 + 32 = 90) and the N number is 32. I will again move the T-Shape down one row.

The total value of this T-Shape is 140 (21 + 22 + 23 + 32 + 42 = 140) and the N number is 42. I will finally move the T-Shape down one row.

The total value of this T-Shape is 190 (31 + 32 + 33 + 42 + 52 = 190) and the N number is 52. I will now analyse these results in a table.

You can see there is a pattern between the numbers. Every time the N goes up 10, the T goes up by 50. I will now do another grid and I predict that when the N goes up by 10 (so it will be 62,) the T will go up by 50 (and be a total of 240.)

The total of this T-Shape is 240 (41 + 42 + 43 + 52 + 62 = 240) and the N number is 62. This proves my prediction and the pattern was correct.

Rule

I am now going to try and find a rule for this question on a 10 by 10 grid, which I am going to change from T into terms of N. The only variable needed after the rule will be N.

Values of T = 1 + 2 + 3 + 12 + 22 = 40 Value of N = 22

Value of T = N + N-10 + N-19 + N-20 + N-21 so

RULE: Value of T = 5N-70

To prove this rule I will test it on a T-Shape with the N number as 92.

5 x 92 - 70 = 390

Value of T = 71 + 72 + 73 + 82 + 92 = 390 Value of N = 92

I have now proved the relationship between the T and N numbers is N x 5 - 70. I am now going to prove the formula works by testing it on a T-Shape I have not yet used and check if the first result wasn’t just a guess.

Value of T = 51 + 52 + 53 + 62 + 72 = 290 Value of N = 72

5 x 72 - 70 = 290

I have now proved the formula works on each ...

This is a preview of the whole essay

Value of T = 51 + 52 + 53 + 62 + 72 = 290 Value of N = 72

5 x 72 - 70 = 290

I have now proved the formula works on each of the two graphs and I will now go onto the next step in my investigation.

Question 2 (Across)

I will now start to answer question 2, what happens when the T-Shape moves across the grid? I will again use the 10 by 10 grid.

The total value of this T-Shape is 40 (1 + 2 + 3 + 12 +22 = 40) and the N number is 22. To see if there is a sequence between the T-Shape, I will move the shape across one column.

The total value of this T-Shape is 45 (2 + 3 + 4 + 13 + 23 = 45) and the N number is 23. I will move the T-Shape across one column again.

The total value of this T-Shape is 50 (3 + 4 + 5 + 14 + 24 = 50) and the N number is 24. I will again move the T-Shape across one row.

The total value of this T-Shape is 55 (4 + 5 + 6 + 15 + 25 = 55). I will now analyse the results of the T-Shape.

You can see there is a pattern by looking at this table. When N goes up 1 the T goes up by 5. I will do one more T-Shape and test a prediction. I predict that when the N number goes up by 1 (so it will be 26,) the T-Total will go up by 5 (and be a total of 60.)

The total value of this T-Shape is 60 (5 + 6 + 7 + 16 + 26 = 60) and the N number is 26. This proves my prediction and the pattern correct.

Rule

I am now going to try and find a rule for a 10 by 10 grid to get from N to T (in terms of N.) This should give me a rule for which the only variable is N.

Value of T = 1 + 2 + 3 + 12 + 22 = 40 Value of N = 22

Value of T = N + N-10 + N-19 + N-20 + N-21 so

RULE: Value of T = 5N-70

To prove my rule I am now going to test it on the T-Shape above (where N = 22.)

5 x 22 - 70 = 40

Value of T = 7 + 8 + 9 + 18 + 28 = 70 Value of N = 28

5 x 28 - 70 = 70

These two graphs have proved that the formula works, and the formula is the same as the one for moving down the 10 by 10 grid. This means that if we know the number of the N, then we can work the out the total of the T on a 10 by 10 grid. I predict that all the formulas for moving down the grid will be the same as the formulas for moving across the grids on each sized grid (e.g. 9 by 9, 8 by 8 and 7 by 7.) I will be proving this prediction in the final stage of these two questions when I combine the formula for the other sized grids.

9 by 9 Grid

Now that I have the formulas for moving the T-Shape across and down on a 10 by 10 grid, I will now apply this knowledge and extend the investigation to a 9 by 9 grid.

Question 2 (Across)

The value of this T-Shape is 37 (1 + 2 + 3 + 11 + 20 = 37) and the N number is 20. I will move the T-Shape across one column again.

The value of this T-Shape is 42 (2 + 3 + 4 + 12 + 21 = 42 and the N number is 21. To continue the sequence, I will move the T-Shape across another column again.

The total of this T-Shape is 47 (3 + 4 + 5 + 13 + 22 = 47) and the N number is 22. I will move the T-Shape across one column one more time.

The total of this T-Shape is 52 (4 + 5 + 6 + 14 + 23 = 52) and the N number is 23. I will now analyse my results of the T-Shape in a table.

You can see there is a pattern by looking at this table. Just like the 10 by 10 grid, when N goes up 1 the T goes up by 5. I will do one more T-Shape and test a prediction. I predict that when the N number goes up by 1 (so it will be 24,) the T-Total will go up by 5 (and be a total of 57.)

Value of T = 5 + 6 + 7 + 15 + 24 = 57 Value of N = 24

This has proved my prediction is correct, so I will now try and find a rule for the grid.

Rule

I am now going to try and find a rule for a 9 by 9 grid to get from N to T (in terms of N.) This should give me a rule for which the only variable is N.

Value of T = 1 + 2 + 3 + 11 + 20 = 37 Value of N = 20

Value of T = N + N-9 + N-17 + N-18 + N-19

RULE: 5N - 63

To prove my rule I am now going to test it on the T-Shape above (where N = 20.)

5 x 20 - 63 = 37

I have now proved the relationship between the T and N numbers is N x 5 - 63. I am now going to prove the formula works by testing it on a T-Shape I have not yet used and check if the first result wasn’t just a guess.

Value of T = 7 + 8 + 9 + 17 + 26 = 67 Value of N = 26

5 x 26 - 63 = 67

This now proves that the formula N x 5 - 63 will work for any T-Number moving across on a 9 by 9 grid.

Question 1 (Down)

I am now going to apply the formula above (N x 5 - 63) for moving down the 9 by 9 grid. I am going to use the N number 74 to test my formula.

Value of T = 55 + 56 + 57 + 65 + 74 = 307 Value of N = 74

I will now use the formula used for moving across the grid and see if it works when moving down the grid.

Value of N = 74

Value of T = 5 x 74 - 63 = 307

This proves the formula for moving across the grid is the same as moving down the grid. I have proved one half of my prediction correct in that the 9 by 9 grid formula is the same and I will now try to prove this on the 8 by 8 grid.

8 by 8 Grid

I am now going to do the 8 by 8 grid for the first 2 questions on the investigation.

Question 2 (Across)

The total of this T-Shape is 34 (1 + 2 + 3 + 10 + 18 = 34) and the N number is 18. I will move my T-Shape across one column.

The total of this T-Shape is 39 (2 + 3 + 4 + 11 + 19 = 39) and the N number is 19. I will again move the T-Shape across one column.

The total of this T-Shape is 44 (3 + 4 + 5 + 12 + 20 = 44) and the N number is 20. I will finally move the T-Shape across one column again.

The total of this T-Shape is 49 (4 + 5 + 6 + 13 + 21 = 49) and the N number is 21. I will now analyse my results of the T-Shape in a small table.

You can see there is a pattern by looking at this table. Just like the 10 by 10 grid and 9 by 9 grid, when N goes up 1 the T goes up by 5. I will do one more T-Shape and test a prediction. I predict that when the N number goes up by 1 (so it will be 22,) the T-Total will go up by 5 (and be a total of 54.)

The total of this T-Shape is 54 (5 + 6 + 7 + 14 + 22 = 54) using the N number 22. This has proved my prediction is correct and I will now try and find a rule for my grid.

Rule

I am now going to try and find a rule for a 8 by 8 grid to get from N to T (in terms of N.) This should give me a rule for which the only variable is N.

Value of T = 1 + 2 + 3 + 10 + 18 = 34 Value of N = 18

Value of T = N + N-8 + N-15 + N-16 + N-17 so

RULE: 5N - 56

To prove my rule I am now going to test it on the T-Shape above (where N = 18.)

5 x 18 - 56 = 34

I have now proved the relationship between the T and N numbers is N x 5 - 56. I am now going to prove the formula works by testing it on a T-Shape I have not yet used and check if the first result wasn’t just a guess.

Value of T = 10 + 11 + 12 + 19 + 27 = 79 Value of N = 27

5 x 27 - 56 = 79

This now proves that the formula N x 5 - 56 will work for any T-Number moving across on a 8 by 8 grid.

Question 1 (Down)

I am now going to apply the formula above (5 x N - 56) for moving down the 8 by 8 grid. I am going to use the N number 58 to test my formula.

Value of T = 41 + 42 + 43 + 50 + 58 = 234 Value of N = 58

I will now use the formula from moving the T-Shape across the 8 by 8 grid to see if it works when moving down the 8 by 8 grid.

5 x 58 - 56 = 234

This proves the formula for moving across the grid is the same as moving down the grid. I have proved my predictions in that the 9 by 9 grid formula is the same for moving up and down and that the 8 by 8 grid formula is the same for moving up and down the grid.

Rule between all Grid Sizes

I will now look at working out a rule for the relationship between N and T for any grid size. To do this I will take a T-Shape for a 9 by 9 grid and try to write in terms of N and G.

Value of T = 1 + 2 + 3 + 11 + 20 = 37 Value of N = 20

This is the T-Shape I will use to find a rule between N and T for any grid size. The N number I am using is 20.

G = Grid Size

I will now put this T-Shape into a formula.

Value of T = N + N-G + N-2G-1 + N-2G + N-2G+1

RULE: 5N-7G + 1-1

I don’t need to keep the +1 and -1 as they both cancel each other out.

So for the grid size of 9 by 9 and the T-Number of 20:

Value of T = 1 + 2 + 3 + 11 + 20 = 37 Value of N = 20

(5 x 20) - (7 x 9) = 37

This proves that the formula works for 9 by 9 grids. I will test it one more time on a different size grid to make sure it was not a coincidence and that it works for all sized grids.

I will use the T-Number 45 and the grid size 8 (8 by 8 grid.) I predict that the total of the formula using these numbers will be 169.

(5 x 45) - (7 x 8) = 169

I will now apply this and draw the T-Shape out to see if the formula works.

Value of T = 28 + 29 + 30 + 37 + 45 = 169 Value of N = 45

This proves my formula works for any grid size and any T-Number.

FORMULA = 5N -7G

Question 3) Rotating the T-Shape

180° Clockwise

Next I will try to find a rule for when the T-shape is this way up (180° clockwise):

Because of the knowledge of the last rule (5N - 7G,) I can move straight on to finding a relationship between N and T for any grid size. To do this I will take a T-Shape from a 9 by 9 Grid and try to write in terms of N and G.

Value of T = 2 + 11 + 19 + 20 + 21 = 73 Value of N = 2 so the rule will be:

Value of T = N + N+G + N+2G + N+2G+1 + N+2G-1

Value of T = 5N + 7G + 1-1 (again the +1-1 cancel out so)

RULE: = 5N + 7G

For this T-Shape N = 2 and G = 9 so:

Value of T = 5 x 2 + 7 x 9 = 73

The rule seems to work on this grid and T-Number so I will now test the rule on a different grid.

Test the Rule

To test this rule I am going to pick a random T-Shape from an 8 by 8 grid and apply the rule to it.

Value of T = 4 + 12 + 19 + 20 + 21 = 76 Value of N = 4

Using the rule 5N + 7G (where N = 4 and G = 8):

Value of T = 5 x 4 + 7 x 8 = 76

The rule T = 5N + 7G is correct for a grid of any size with a T-Shape rotated 180° clockwise.

90° Clockwise

I am now going to try and find a formula for a T-Shape facing this way up (90° Clockwise):

Again, because of the knowledge of the previous rules I can move straight on to finding a relationship between N and T for any grid size. To do this I will take a T-Shape from a 9 by 9 Grid and try to write in terms of N and G.

Value of T = 3 + 10 + 11 + 12 + 21 = 57 Value of N = 10

so the rule will be:

Value of T = N + N+1 + N+2 + N+2+G + N+2-G

Value of T = 5N + 7 + G–G (the +G-G cancel each other out so):

RULE: = 5N + 7

For this shape N = 10 so:

Value of T = 5 x 10 + 7 = 57

The rule seems to work on this grid and T-Number so I will now test the rule on a different grid.

Test the Rule

To test this rule I am going to pick a random T-Shape from an 8 by 8 grid and apply the rule to it.

Value of T = 3 + 9 + 10 + 11 + 19 = 52 Value of N = 9

Using the rule 5N + 7 (where N = 9):

Value of T = 5 x 9 + 7 = 52

The rule T = 5N + 7 is correct for a grid of any size with a T-Shape rotated 90° clockwise.

270° Clockwise

I am now going to try and find a formula for a T-Shape facing this way up (270° Clockwise):

Because of the knowledge of the previous rules I can again move straight on to finding a relationship between N and T for any grid size. To do this I will take a T-Shape from a 9 by 9 Grid and try to write in terms of N and G.

Value of T = 1 + 10 + 11 + 12 + 19 = 53 Value of N = 12

so the rule is:

Value of T = N + N-1 + N-2 + N-2-G + N-2+6

Value of T = 5N – 7 + G–G (the +G-G cancel each other out so):

RULE = 5N – 7

For this shape N = 12 so:

Value of T = 12 x 5 - 7 = 53

The rule seems to work on this grid and T-Number so I will now test the rule on a different grid.

Test the Rule

To test this rule I am going to pick a random T-Shape from an 8 by 8 grid and apply the rule to it.

Value of T = 1 + 9 + 10 + 11 + 17 = 48 Value of N = 11

Using the rule 5N - 7 (where N = 11):

Value of T = 5 x 11 - 7 = 48

The rule T = 5N - 7 is correct for a grid of any size with a T-Shape rotated 270° clockwise.

Summary of Work so far

So far in this T-Shapes investigation I have found out rules to work out the T-Total when the T-Shape is on any grid size and in any orientation from the T-Number (e.g. rotated, moved down or across.)

I have found out the rules for moving the T-Shapes across and down a grid on different grid dimensions. These are the rules I have found out:

Value of T = 5N - 70 (10 by 10 grid moving across)

Value of T = 5N - 70 (10 by 10 grid moving down)

Value of T = 5N - 63 (9 by 9 grid moving across)

Value of T = 5N - 63 (9 by 9 grid moving down)

Value of T = 5N - 56 (8 by 8 grid moving across)

Value of T = 5N - 56 (8 by 8 grid moving down)

Value of T = 5N - 7G (Rule between all grid sizes)

The rules I have found for rotating the T-Shapes on different grid dimensions are:

Value of T = 5N – 7G (Stationary, facing up)

Value of T = 5N + 7G (180° Clockwise)

Value of T = 5N + 7 (90° Clockwise)

Value of T = 5N – 7 (270° Clockwise)

Question 4) Smallest Value of the T-Shape in the Number Grid

I am now onto my second to last question in this investigation. In this stage I am going to try and find the smallest value in all the number grids; 10 by 10, 9 by 9 and 8 by 8. I will begin by trying to find the smallest value on the 10 by 10 grid. To do this I will rotate my T-Shape 90°, 180° and 270° to see which has the smallest value instead of just taking a guess at the first shape.

10 by 10 Grid

I will first start with my T-Shape being stationary like in question 1:

The total value of this T-Shape is 40 (1 + 2 + 3 + 12 + 22 = 40) and the N number is 22.

I will now rotate the T-Shape 90° clockwise:

The total value of this T-Shape is 62 (3 + 11 + 12 + 13 + 23 = 62) and the N number is 11.

I will now rotate the T-Shape another 90° clockwise, so that the T-Shape has moved around a total of 180° clockwise.

The total value of this T-Shape is 80 (2 + 12 + 21 + 22 + 23 = 80) and the N number is 2.

I will finally rotate the T-Shape another 90° clockwise, so that the T-Shape has moved around a total of 270° clockwise.

The total value of this T-Shape is 58 (1 + 11 + 12 + 13 + 21 = 58) and the N number is 13.

I now know that the smallest value on a 10 by 10 grid is 40, where the N number is 22 and the T-Shape is stationary.

Rule

I am now going to try and find a rule for the largest value of the T-Shape on a 10 by 10 grid.

G = Grid

N = 22

G = 10

To get from G to N, you need to times it by 2 and then add 2 on to that to get the total of 22. So:

RULE: 2G + 2

9 by 9 Grid

I will first start with my T-Shape being stationary like the first T-Shape on the 10 by 10 grid.

The value of this T-Shape is 37 (1 + 2 + 3 + 11 + 20 = 37) and the N number is 20.

I will now rotate the T-Shape 90° clockwise:

The value of this T-Shape is 57 (3 + 10 + 11 + 12 + 21 = 57) and the N number is 21.

I will again rotate the T-Shape another 90°, so that the T-Shape will have rotated 180° altogether.

The value of this T-Shape is 73 (2 + 11 + 19 + 20 + 21 = 73) and the N number is 2.

I will finally rotate the T-Shape another 90° again, so that the T-Shape will have rotated a total amount of 270°.

The value of this T-Shape is 53 (1 + 10 + 11 + 12 + 19 = 53) and the N number is 12.

I now know that the smallest value on a 9 by 9 grid is 37, where the N number is 20 and the T-Shape is stationary. I have also found out that on the 10 by 10 and 9 by 9 grid, the smallest value is when the T-Shape has been stationary, so I predict that for the 8 by 8 grid, the smallest value will be when the T-Shape is stationary.

I will now use the rule 2G + 2 to see if the rule works. I will use the G (Grid) as 9 and N as 20.

So to get from G = 9 to N = 20 using the rule you times 9 by 2 to get 18, then add on 2 which gives 20. The rule works, however, I will still try the rule again in the 8 by 8 grid to prove it correct.

8 by 8 grid

I will first start with my T-Shape being stationary like the first T-Shape on the 10 by 10 and 9 by 9 grid.

The total of this T-Shape is 34 (1 + 2 + 3 + 10 + 18 = 34) and the N number is 18.

I will now rotate the T-Shape by 90° clockwise:

The total of this T-Shape is 52 (3 + 9 + 10 + 11 + 19 = 52) and the N number is 9.

I will rotate the T-Shape another 90° clockwise, and the T-Shape will have rotated 180° clockwise altogether.

The total of this T-Shape is 66 (2 + 10 + 17 + 18 + 19 = ) and the N number is 2.

I will finally rotate the T-Shape one more time at 90°, totalling a rotation of 270°.

The total of this T-Shape is 48 (1 + 9 + 10 + 11 + 17 = 48) and the N number is 11.

I now know that the smallest value on a 8 by 8 grid is 34, where the N number is 18 and the T-Shape is stationary.

I have now proved my prediction correct in which the T-Shape will be the smallest value when it is stationary on a 8 by 8 grid. This is the same for both the 10 by 10 and 9 by 9 grid.

I will now try and prove the rule for the last time on the 8 by 8 grid. I will use the G (Grid) as 8 and the N as 18.

So to get from G = 8 to N = 18 you times 8 by 2 which equals 16, and then add 2 which equals 18. This proves the rule is correct for the smallest value on a T-Shape.

Question 5) Largest Value of the T-Shape in the Number Grid

10 by 10 grid