Aim 1- the solution
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
T69= 50+51+52+60+69
=282
T22=3+4+5+13+22
=47
In the diagram below it shows the difference between the T-number and the other numbers. First is the T-shape in question:
1 2 3
11
20
This is the T-shape and here is the Difference T-shape:
N-19 N-18 N-17
N-9
N
This shows the difference N= T-number
In the T on the previous page I have noticed that the first difference from N is 9 which is also the Width of the square.
I´ll put that idea into another T. Note W= width number(9)
N-(2W-1) N-2W N-(2W+1)
N-W
N
This is the same thing as before but shown algebraically.
The formula for the Value of the T-total now is shown as:
5N-7W=T-total
Aim 2- different sizes and relationship
I know this works for the grid 9 by 9 but I´m not sure if it´ll work for any other grids.
Here is a test for a 10 by 10 grid
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
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 is 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.
