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
To prove that this works, take any number, for example t – number = 47.
5 * 47 = 235. 235 – 7 * 9 = 63
235 – 63 = 172 = t – total
28 + 29 + 30 + 38 + 47 = 172
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.
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.
As before the numbers that make up the ‘T’ can be written in terms of the T – number.
On an 8 by 8 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
Keeping the same T – number…
N-17 N-16 N-15
N-8
N
Simplified this can be written as 5N - 56
3 + 4 + 5 + 12 + 20 = 44
5 * 20 –56 = 44
This proves that the formula works !
In an algebraic formula.
N-2W+1 N-2W N-2W-1
N-W
N
This gives 5N – 7W = T-total. The same as before
5N – 63 9 by 9 grid 5N – 56 8 by 8 grid
63 – 56 = 7
5N – 7W = T-total : This formula will work for any size grid so long as the T is in its upright position.
10 by 10 grid 5N – 70 9 by 9 grid 5N – 63 8 by 8 grid 5N – 56 7 by 7 grid 5N - 49 and so on …
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
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+(4*36)=216
216 / 12 = 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.
Below is a sheet with more simple things on it satisfying the same aim roughly explaining the above.
The formula 5N – 7W will only work for T’s in the upright position. Eg 1 2 3
11
20
It will not work for a T rotated 90 degrees clockwise. To prove that this is the case…
3
10 11 12 5 * 10 – 7 * 9 = -13
21
10 + 11+ 12 + 3 +21 = 57
Below is a table of results which help me to find out formulas and to see if there are any obvious trends.
As you can see from the results the third difference of the difference is an inverse of the first difference of the difference.
So the formula for a T that has been rotated by 180 degrees should be an inverse of the formula used to work out a T in its upright position. Therefore
5N + 7W
It’s a plus sign as it’s the inverse of a minus sign.
Proof …
8 by 8 grid
1 2 3 4 5 6 7 8 (5 * 2) + (7 * 8) = 66
9 10 11 12 13 14 15 16 2 + 10 +17 +18 +19 = 66
17 18 19 20 21 22 23 24 It works !
25 26 27 28 29 30 31 32
However the formula will only work for T’s rotated by 180 degrees.
In terms of the T – number the 270 degree rotation is equal to :
5N – 7
I found this out by accident I found out that it worked for one and then I tried it on other size grids and it worked.
5 * 21 – 7 = 98
21 + 22 + 23 + 13 + 29 = 98
see it works. I then tested it on other places and it still worked. It wasn’t just a coincidence.
I took this further and used the same method as before in working out the inverse of this. To find out the T – total for a T that has been rotated by 90 degrees it will be the inverse of the 270 degree one.
5 * 14 + 7 = 77
14 + 15 + 16 + 24 + 8 =77 This also works.
5N + 7
5N - 7
5N – 7W
5N + 7W
