T22=1+2+3+12+22
=42 I notice this is 5 more than 9 by 9
T69=48+49+50+59+69
=275 obviously no pattern there.
Method test?
(695)-70=275 YES it worked
My method seems to have worked out as it is logical and fairly straight forward to explain.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
As there are 10 in each row it’s obvious that the row above will be 10 less than the row below. So 68 are 10 less than the T-number 78. If you calculate the whole T you realise that row 2 is 10 less than row 1 and row 3 is 20 less than row 1, but there are three relevant numbers in row 3 which are 19 less and 21 less than the T-number. These cancel out to form 20 each, So finally we get (110)+(610)=(710)
=70
Or =7W
12 by 12
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96
97 98 99 100 101 102 103 104 105 106 107 108
109 110 111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130 131 132
133 134 135 136 137 138 139 140 141 142 143 144
T62=37+38+39+50+62 also = (625)-(712)
= 226 =226
T141=116+117+118+129+141 also = (1415)-(712)
=621 =621
Aim3- Transformations stretches and there effects on the formula
I’ll do this with a 12 by 12 first, as this will give me enough accuracy to start with.
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70 71 72
73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96
97 98 99 100 101 102 103 104 105 106 107 108
109 110 111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130 131 132
133 134 135 136 137 138 139 140 141 142 143 144
Stretch A will be called ST64 as it starts at 64; it’s a stretch of 2 in both directions.
St64=26+27+28+29+30+40+52+64
=296
I think I can work out the formula using my previous method so:
12+24+36+ (436) =216
21612=18
this means the formula is:
8N-18W=T-total
8N= number of integers in the T-shape
18W=difference number calculated
Conclusion:
The size of the T-shape calculates the number before N in the formula and the grid size calculates the value of W. the number before W is calculated by looking at the rows and finding how many rows away from the T-number they are. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.
T- Total
Look at the T-shape drawn on the 9 by 9 number grid.
The total of the numbers inside the T-shape is 1+2+3+11+20=37
This is called the T-total.
The number at the bottom of the T-shape is called theT-number.
The T-number for this T-shape is 20.
If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.
You will notice that the centre column of the T-Shape is going up in 9's because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:-
T-total = T-19+T-18+T-17+T-9+T
= 5T-63
Now to test this formula to see if it works
For T-Total I will use the letter X
For the T-Number I will use the letter T
So X = 5T-63
T = 20
X = 5x20-63
= 100-63
= 37
Now I will do two more to check to see if it will work anywhere on the grid.
T=43
X=5x43-63 24+25+26+34+43=152
X=215-63
X=152
T=49
X=5x49-63 30+31+32+40+49=182
X=245-63
X=182
I have tested the formula for three different T-numbers and the formula works for a 9 by 9 grid.
The full formula for this size grid is:- X=5T-63
When trying to fine out the T-Total in a 6 by 6 number grid,
If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.
This time the centre column of the T-Shape is going up in 6's because of the table size. The formula for this size grid is:-
T-Total = T-13+T-12+T-11+T-6+T
= 5T-42
Now to test this formula to see if it works, again
For the T-total I will use the letter X
For the T-Number I will use the letter T
So X = 5T-42
T = 14
X = 5x14-42
= 70-42
= 28
Now I will do two more to check to see if it will work anywhere on the grid.
T=22
X=5x22-42 9+10+11+16+22=68
=110-42
= 68
T=35
X=5x35-42 22+23+24+29+35=133
=175-42
=133
I have tested the formula for three different T-numbers and the formula works for a 6 by 6 grid.
The full formula for this size grid is:- X=5T-42
Below is a grid with parts of the formula in. I will use this to
figure out a formula for any T-Total on any width grid.
T-total = T-2G-1
T-2G
T-2G+1
T-G
T
X=5T-7G
If the Grid Width (= G), and the T-Number (= T), the T-Total (= X) is calculated by
X=5T-7G
Now I'm going to try the grid that I drew earlier i.e. G=9
The formula is X=5T-7G
Which changes to X=5T-7x9
X=5T-63
I will now try the formula on a 7 by 7 grid.
X=5X17-49 (2+3+4+10+17=36
X=85-49
X=36
I am now going to try this formula on a 3 by 5 grid.
X=5X8-21 (1+2+3+5+8=19
X=40-21
X=19
I am now going to try this formula on a 6 by 10 grid.
X=5X33-42 (20+21+22+27+33=123
X=165-42
X=123
Now I have worked the formula out for the T-Shape in the one position, I am going to translate the T-Shape to different positions and investigate the relationships between the T-Total, the T-number and the grid size.
The other T-Shape's that I can find a formula's are shown below:-
A 3 by 3 grid can be used for the other 3.
T
T+G
T+2G
T+2G-1
T+2G+1
7G+5T
X=7G+5T=7x3+5x2
=21+10 (2+5+7+8+9=31)
=31
I will now try this new formula on a 4 by 7 grid.
7G+5T
x=7x4+5x10 (17+18+19+14+10=78)
=28+50
=78
I will now try it on a 8 by 3 grid.
7G+5T
x=7x9+5x2 (19+20+21+11+2=73)
=56+10
=73
I have tried this new formula for an inverted T and it works, the formula is: 7G+5T
T
T-1
T-2
T-2-G
T-2+G
5T-7
X= 5T-7=5x6-7
=30-7 (1+4+7+5+6=23)
=23
I will now try the new formula on a 4 by 4 grid
5T-7
X=5x11-7 (5+9+13+10+11=48)
=48
And on a 4 by 9 grid.
5T-7
x=5x19-7 (13+17+21+18+19=88)
=88
I have tested this new formula for a side ways T the formula works it is: 5T-7
T
T+1
T+2
T+2+G
T+2-G
5t+7
The formula is 5T+7
X=5T+7=5x4+7
=20+7 (3+6+9+5+4=27)
=27
I will now try the new formula on a 6 by 6 grid.
5T+7
X=5x15+7 (11+17+23+16+15=82)
=82
And on a 11 by 5 grid.
5T+7
X=5x28+7 (28+29+30+19+41=147)
=147
I have tested this new formula for a side ways T the formula works it is: 5T+7
The formula for a this way up is: X=5T-7G
The formula for a this way up is: X=7G+5T
The formula for a this way up is:X=5T+7
The formula for a this way up is: X=5T-7
____________________________________________________
I am now going to investigate the effect on the T-Total when the T-Shape is transformed, rotated 90°, both T-Shapes will have the same T-Number like this:
T- Number = 12
Purple T-Total = 25
Grey T-Total = 67 a difference of 42….. The formula 5T - 7G + 42
Now I'm going to see what happens on a 6 by 6 width Grid
T- Number = 14
Purple T-Total = 33
Grey T-Total = 82 a difference of 49….. The formula 5T - 7G + 49
7 by 7 width Grid
T- Number = 16
Purple T-Total = 41
Grey T-Total = 97 a difference of 56….. The formula 5T - 7G + 56
Table of Results
_________________________________________________________________________________
Now I will rotate it again 180° from it's original place
T- Number = 12
Purple T-Total = 25
Grey T-Total = 95 a difference of 70….. The formula 5T - 7G + 70
Now I'm going to see what happens on a 6 by 6 width Grid
T- Number = 14
Purple T = 28
Grey T = 112 a difference of 84….. The formula 5T - 7G + 84
7 by 7 Grid
T- Number = 16
Purple T-Total = 31
Grey T-Total = 129 a difference of 98….The formula 5T - 7G + 98