'T' Totals Investigation.
'T' Totals Investigation
Introduction
We have been given the task to find the relationship between the 'T' ? and the 'T' Total 9x9 grid. The 'T' ? is the number at the bottom of the T shape and the 'T' Total is the sum of all the numbers inside the T.
Example
'T' Total = 1+2+3+11+20
= 37
To find the relationship between the 'T' ? and the 'T' Total, algebra will be needed. Therefore the 'T' ? will be represented by 'X' and the 'T' Total will be represented by 'T'. I will substitute the numbers in the T and replace them with those numbers but with the 'T' ? (X) as the subject.
Example
From this we can come up with a formula. I will check the formula with one of my other results for that grid. If the relationship between the T' ? and the 'T' Total match the formula will be correct and the problem will be solved.
Table of Results for a 9x9 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
20
37
42
47
67
272
80
337
T = X+(X-9)+(X-17)+(X-18)+(X-19)
T = 5X-63
Check
T = 5X-63
T = 5x42-63
T = 147
Formula to find 'T' Total (T) if you have the 'T'? (X) in a 9x9 grid:
=5X-63
Now I have solved the first problem I am going to further my investigation and vary the grid sizes. I am going to use the same method to find the formulas. The formulas should been linked and if I can find that link I cam make a formula that incorporates grid size. This would mean I could find the 'T' Total if I had the T'? in any grid size.
Table of Results for an 8x8 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
8
34
29
89
50
94
61
244
T = X+(X-8)+(X-15)+(X-16)+(X-17)
T = 5X-56
Check
T = 5X-56
T = 5x50-56
T = 194
Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid:
=5X-56
Table of Results for a 10x10 Grid
See next page
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
77
88
89
90
91
92
93
94
95
96
87
89
99
00
'T' ? (X)
'T' Total (T)
22
40
37
15
69
275
75
305
T = X+(X-10)+(X-19)+(X-20)+(X-21)
T = 5X-70
Check
T = 5X-70
T = 5x69-70
T = 275
Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid:
=5X-70
I have now found the formulae for three grid sizes:
8x8 = 5X-56
9x9 = 5X-63
0x10 = 5X-70
The link between these formulae is that the number being subtracted is always the grid size multiplied by 7:
8x8 = 5X- (8x7) = 5X-56
9x9 = 5X- (9x7) = 5X-63
0x10 = 5X- (10x7) = 5X-70
Two other links are:
) The square above the 'T' ? is always the 'T' ? subtracted by the grid size.
2) The square above that is always the 'T' ? subtracted by the grid size which has been multiplied by 2.
8x8
9x9
0x10
Therefore these squares on the grid could be made into X-G and X-2G ('G' meaning grid size):
The last two links are in the 'T' Shapes are:
a) The square to the right of the middle square on the top is always the grid size multiplied by 2 and subtracted by 1.
b) The square to the left of the middle square on the top is always the grid size multiplied by 2 and then 1 is added.
8x8
9x9
0x10
Therefore these squares on the grid could be made into X-2G-1 and X-2G+1:
Now that the 'T' shape has been put into algebra that incorporates the grid size, a formula that incorporates the grid size can be made.
T= X+(X-G)+(X-2G)+(X-2G-1)+(X-2G+1)
T= 5X-7G
Check 8x8 Check 9x9 Check 10x10
T = 5X-7G T = 5X-7G T = 5X-7G
T = 5x50-7x8 T = 5x42-7x9 T = 5x69-7x10
T = 250-56 T = 210-63 T = 345-70
T = 194 T = 147 T = 275
Formula to find 'T' Total if you have the 'T'? (X) and the grid size (G):
=5X-7G
To further the investigation, I will now turn the 'T' shape 90° clockwise about the 'T' ? square so the 'T'? will be in a different place, therefore the ...
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T= 5X-7G
Check 8x8 Check 9x9 Check 10x10
T = 5X-7G T = 5X-7G T = 5X-7G
T = 5x50-7x8 T = 5x42-7x9 T = 5x69-7x10
T = 250-56 T = 210-63 T = 345-70
T = 194 T = 147 T = 275
Formula to find 'T' Total if you have the 'T'? (X) and the grid size (G):
=5X-7G
To further the investigation, I will now turn the 'T' shape 90° clockwise about the 'T' ? square so the 'T'? will be in a different place, therefore the formula will be different. I will alter the grid sizes as well and find a formula for all grid sizes.
Table of Results for an 8x8 Grid
X
See next page
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
7
92
20
07
50
257
54
277
T = X+(X+1)+(X+2)+(X-6)+(X+10)
T = 5X+7
Check
T = 5X+7
T = 5x17+7
T = 92
Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid if the 'T' shape is turned 90° clockwise about the 'T' ? square:
=5X+7
Table of Results for a 9x9 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
0
57
24
27
48
247
70
357
T = X+(X+1)+(X+2)+(X-7)+(X+11)
T = 5X+7
Check
T = 5X+7
T = 5x24+7
T = 127
Formula to find 'T' Total if you have the 'T'? (X) in a 9x9 grid if the 'T' shape is turned 90° clockwise about the 'T' ? square:
=5X+7
Table of Results for a 10x10 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
00
'T' ? (X)
'T' Total (T)
1
57
34
77
72
367
77
392
T = X+(X+1)+(X+2)+(X-8)+(X+12)
T = 5X+7
Check
T = 5X+7
T = 5x34+7
T = 177
Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid if the 'T' shape is turned 90° clockwise about the 'T' ? square:
=5X+7
The obvious link between these formulae is that they are all the same:
8x8 = 5X+7
9x9 = 5X+7
0x10 = 5X+7
Therefore if the 'T' shape is turned 90° clockwise about the 'T' ? square the grid size doesn't matter and the formula is:
=5X+7
I will now turn the 'T' shape 180° clockwise about the 'T' ? square so the 'T'? will be in a different place, therefore the formula will be different. I will alter the grid sizes as well and find a formula for all grid sizes.
Table of Results for an 8x8 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
2
66
6
86
30
206
35
231
T = X+(X+8)+(X+15)+(X+16)+(X+17)
T = 5X+56
Check
T = 5X+56
T = 5x30+7
T = 206
Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid if the 'T' shape is turned 180° clockwise about the 'T' ? square:
=5X+56
Table of Results for a 9x9 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
2
73
23
78
44
283
56
343
T = X+(X+9)+(X+17)+(X+18)+(X+19)
T = 5X+63
Check
T = 5X+7
T = 5x23+63
T = 178
Formula to find 'T' Total if you have the 'T'? (X) in a 9x9 grid if the 'T' shape is turned 180° clockwise about the 'T' ? square:
=5X+63
Table of Results for a 10x10 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
00
'T' ? (X)
'T' Total (T)
2
80
9
60
46
300
62
380
T = X+(X+10)+(X+19)+(X+20)+(X+21)
T = 5X+70
Check
T = 5X+70
T = 5x19+70
T = 160
Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid if the 'T' shape is turned 180° clockwise about the 'T' ? square:
=5X+70
The link between these formulae is that the number being added is always the grid size multiplied by 7. This is the same as the formula for the normal 'T' shape:
8x8 = 5X+ (8x7) = 5X+56
9x9 = 5X+ (9x7) = 5X+63
0x10 = 5X+ (10x7) = 5X+70
Two other links are:
) The square below the 'T' ? is always the 'T' ? with the grid size added to it.
2) The square below that is always the 'T' ? added to the grid size which has been multiplied by 2.
8x8
9x9
0x10
Therefore these squares on the grid could be made into X+G and X+2G:
This is the same as the formula for the normal 'T' shape except it is addition instead of subtraction.
The last two links are in the 'T' Shapes are:
a) The square to the right of the middle square on the bottom is always the grid size multiplied by 2 and subtracted by 1.
b) The square to the left of the middle square on the bottom is always the grid size multiplied by 2 and then 1 is added.
8x8
9x9
0x10
Therefore these squares on the grid could be made into X-2G-1 and X-2G+1:
Now that the 'T' shape has been put into algebra that incorporates the grid size, a formula that incorporates the grid size can be made.
T= X+(X+G)+(X+2G)+(X+2G-1)+(X+2G+1)
T= 5X+7G
Check 8x8 Check 9x9 Check 10x10
T = 5X+7G T = 5X+7G T = 5X+7G
T = 5x2+7x8 T = 5x2+7x9 T = 5x2+7x10
T = 10+56 T = 10+63 T = 10+70
T = 66 T = 73 T = 80
Formula to find 'T' Total if you have the 'T'? (X) and the grid size (G) and the 'T' shape is turned 180° clockwise about the 'T' ? square :
=5X+7G
This formula is the similar to the normal 'T' shape except it is an addition instead of a subtraction.
I will now turn the 'T' shape 270° clockwise about the 'T' ? square so the 'T'? will be in a different place, therefore the formula will be different. I will alter the grid sizes as well and find a formula for all grid sizes.
Table of Results for an 8x8 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
1
48
30
43
44
213
56
273
T = X+(X-1)+(X-2)+(X+6)+(X-10)
T = 5X-7
Check
T = 5X-7
T = 5x30-7
T = 143
Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid if the 'T' shape is turned 270° clockwise about the 'T' ? square:
=5X-7
Table of Results for a 9x9 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
2
53
43
208
58
283
72
353
T = X+(X-1)+(X-2)+(X+7)+(X-11)
T = 5X-7
Check
T = 5X-7
T = 5x43-7
T = 208
Formula to find 'T' Total if you have the 'T'? (X) in a 9x9 grid if the 'T' shape is turned 180° clockwise about the 'T' ? square:
=5X-7
Table of Results for a 10x10 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
00
'T' ? (X)
'T' Total (T)
3
58
38
83
55
268
89
437
T = X+(X-1)+(X-2)+(X+8)+(X-12)
T = 5X-7
Check
T = 5X-7
T = 5x55-7
T = 268
Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid if the 'T' shape is turned 180° clockwise about the 'T' ? square:
=5X-7
The obvious link between these formulae is that they are all the same:
8x8 = 5X-7
9x9 = 5X-7
0x10 = 5X-7
Therefore if the 'T' shape is turned 270° clockwise about the 'T' ? square the grid size doesn't matter and the formula is:
=5X-7
This formula is similar to the formula for the 'T' shape that is turned 90° about the 'T' ? square except it is a subtraction instead of an addition.
To further my investigation even further I am going to explore vectors. I will vary grid size as before and see if I can find a formula that incorporates vectors, grid size and the 'T'?. I will solve this by an algebraic method
Table of Results for an 8x8 Grid
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
8
34
(1,-2)
35
19
30
94
(-1,1)
21
49
55
219
(-5,-1)
58
234
59
239
(2,-2)
45
69
From this you can see:
. Every time it moves one to the right, 1 is added to the 'T' ?. Because there are 5 squares in a 'T' shape every time it is moved one to the right the 'T' Total will increase by 5.
2. Every time it moves one down, 8 is added to the 'T' ?. 8 is the grid size so every time it moves down the 'T' total will increase by the grid size multiplied by the number of squares. In this case it is 8x5 which is 40; this means every time the 'T' shape moves down by one the 'T' total will increase by 40
Check (1)
Check (2)
Now I to see if this is correct for all grid sizes I will test it against a 9x9 grid and a 10x10 grid. For check 2 the 'T' total for 9x9 should go up by 45 (5x9) and for 10x10 it should go up by 50 (5x10):
Check for 9x9 (1)
Check for 9x9 (2)
Check for 10x10 (1)
Check for 10x10 (2)
The 'T' shape has been moved on the vector of (1,-1). From this we can make a formula to find the 'T' total of the new 'T' shape. For this we shall call the coordinates (A,B). As we have already seen the original 'T' Total has 5 added to it if it moves one to the right and the grid size multiplied by 5 if it moves down. As 'A' represents horizontal movement and 'B' vertical movement then we will have somewhere in the equation 5A and 5GB ('G' standing for grid size). The 5A would be added to the original 'T' total because it is increasing along the x axis and GB would be subtracted because it is decreasing on the y axis. With this information we can make the equation:
T + 5A - 5GB
However if we can make it simpler by putting brackets into the equation. This means we can eliminate one of the fives to put it in its simplest form. This would mean our final equation would be:
T + 5(A-GB)
I will test this formula on 8x8, 9x9 and 10x10 grids. I will also make 'A' negative and see if the formula still works which it should do.
Check 8x8
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
55
219
(-5,-1)
58
234
59
239
(2,-2)
45
69
T1 = T+5(A-GB)
T1 = 219+5(-5-8x-1)
T1 = 219+5x3
T1 = 219+15
T1 = 234
T1 = T+5(A-GB)
T1 = 239+5(2-8x2)
T1 = 239+5x-14
T1 = 239-70
T1 = 169
Check for 9x9
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
26
67
(-2,-4)
60
237
48
77
(-1,3)
20
37
T1 = T+5(A-GB)
T1 = 177+5(-1-9x3)
T1 = 177+5x-28
T1 = 177-140
T1 = 37
T1 = T+5(A-GB)
T1 = 67+5(-2-9x-4)
T1 = 67+5x34
T1 = 67+170
T1 = 237
Check for 10x10
X
See next page
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
00
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
39
25
(0,-2)
59
225
64
250
(3,0)
67
265
T1 = T+5(A-GB)
T1 = 125+5(0-10x-2)
T1 = 125+5x20
T1 = 125+100
T1 = 225
T1 = T+5(A-GB)
T1 = 250+5(3-10x0)
T1 = 250+5x3
T1 = 250+15
T1 = 265
From these tests I can successfully say that the formula to find a new 'T' total if you know the original 'T' total, the grid size and the vector it has been moved on is:
T + 5(A-GB)
I will now investigate if this formula works if the 'T' shapes have been turned 90° clockwise, 180° clockwise and 270° clockwise about the 'T' ? square. I will use different grid sizes and different vectors to make the test fair.
Table of Results for an 8x8 Grid if 'T' shape has been turned 90° clockwise about the 'T' ? square
X
See next page
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
20
07
(-2,1)
0
57
50
257
(4,1)
46
237
T1 = T+5(A-GB)
T1 = 107+5(-2-8x1)
T1 = 107+5x-10
T1 = 107-50
T1 = 57
T1 = T+5(A-GB)
T1 = 257+5(4-8x1)
T1 = 257+5x-4
T1 = 257-20
T1 = 237
Table of Results for a 9x9 Grid if 'T' shape has been turned 90° clockwise about the 'T' ? square
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
24
27
(-4,-2)
38
97
48
247
(3,-2)
69
352
T1 = T+5(A-GB)
T1 = 127+5(-4-9x-2)
T1 = 127+5x14
T1 = 127+70
T1 = 197
T1 = T+5(A-GB)
T1 = 247+5(3-9x-2)
T1 = 247+5x21
T1 = 247+105
T1 = 352
Table of Results for a 10x10 Grid if 'T' shape has been turned 90° clockwise about the 'T' ? square
X
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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
00
'T' ? (X)
'T' Total (T)
Vector
New 'T' Number (X1)
New 'T' Total (T1)
1
62
(2,-4)
53
272
77
392
(0,5)
27
42
T1 = T+5(A-GB)
T1 = 62+5(2-10x-4)
T1 = 62+5x42
T1 = 62+210
T1 = 272
T1 = T+5(A-GB)
T1 = 392+5(0-10x5)
T1 = 392+5x-50
T1 = 392-250
T1 = 142
Through these tests I can see that the formula does work if the 'T' shape is turned 90° clockwise about the 'T' ? square.
Tom Ballard
