Investigation into T-shapes.

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Daniel Turner

Investigation into T-shapes

Looking at the 9-9 grid below and the T-shape drawn on it,

The total number of the numbers on the inside of the T-shape is called the T-total

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

The t-total for this T-shape is:
1+2+3+11+20=37
so 37 = T-total

The number at the bottom is the T-number, so the T-number for this shape is 20

Aims:
1) Investigate the relationship between the T-total and the T-number
2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size.
3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.


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

Join now!

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 ...

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