Contents:
Introduction. T-total, T-number…………………………….………………………3
Methods……………………………………………………………………………..5
Evaluation of Results……………………………………………………………….8
Introduction. T-total, T-number.
This coursework is about trying to find a connection between the t-totals and t-number according to the t-shape
Here is an example of a T-shape drawn on a 9 x 9 number grid
The total of all the numbers in the t-shape is 20+11+1+2+3=37.
This is called the t-total.
The number at the bottom of the t-shape (20) is called the t-number. In this project, I will try to investigate the relationship between the T-total and the T-number.
Method
During the start of the coursework, I attempted to select various t-numbers from a 9 x
9 number grid and calculated into t-totals.
T-number: T-total
- 22+13+3+4+5=47
44 44+35+25+26+27=157
89 89+80+70+71+72=391
Then I used the T-number to work out the rule by subtracting the T-number with the other minor numbers inside the T-shape;
This equals 5n-63
Then I calculated an overall rule that shows a relationship
Between the T-total and the T-number alongside the grid number.
This is the possible rule that matches along with any grid number that shows the t-shape
Below I apply same rule to the 6, 7 and 8 grids:
T-number: T-total:
22 22+16+9+10+11=68
44 44+38+31+32+33=178
80 80+74+567+68+69=358
89 89+83+76+77+78=403
Therefore, the rule is 5n-42
T-number: T-total:
- 24+17+9+10+11=71
55 55+48+40+41+42=226
65 65+58+50+51+52=276
104 104+97+89+90+91=471
Therefore, the rule is 5n-49
T-number: T-Total:
26 26+18+9+10+11=74
23 23+15+6+7+8=58
54 54+46+37+38+39=214
74 74+66+57+58+59=284
Therefore, the rule is 5n-56
Evaluation of Results
1) General rule findings
The above results are generalized in formulas (rules) for each grid size and presented in a following table:
By observation of the above table the general rule applied on the above grid sizes is then: 5n-7g where g=6 to 9.
2) Interrelation between T-totals and Translation vectors
By observation between the T-totals and translation vectors in a typical 9-size grid it appears that the T-totals are directly connected with the T-shape moving sideways and vertically in the grid as follows:
Examples in the following vectors
1 (moving T-shape by one grid horizontally to the right) the T-totals increase by 5
0 each time
-1 (moving T-shape by one grid horizontally to the left) the T-totals decrease by 5
0 each time
0 (moving T-shape by one grid vertically down) the T-totals increase by 45 each time
1
0 (moving T-shape by one grid vertically down) the T-totals decrease by 45
-1 each time.
Similarly by moving as above horizontally or vertically the T-shape by two grids (to the right or down) as follows:
2 T-total increase by 5x2=10
0
0 T-total increase by 45x2=90
2
3) Analysis of observation and general rule findings:
The T-totals increase when shifting the T-shape by 3, 4 etc grids to the right or down is by 5x or 45y respectively. Similarly values as above for decrease of T-totals when shifting grids to the left or upwards.
General Rule finding:
From the above observations a general rule may be applied as follows:
0 T-total increase by 45y
y
Similarly
x T-total increase by 5x
0
Here is an example of the method of translation vectors that I have applied:
T-number: T:total:
30 30+21+11+12+13=87
1 t: number=31
0 T-total=31+22+12+13+14=92
2 t-number=32
0 T-total=32+23+13+14+1=97
0 t-number=39
1 T-total=39+30+21+20+22=132
0 t-number=48
2 T-total=48+39+29+30+31=177
By using of the rule 5n-63 in the translation vectors the t-totals calculated as follows for example in the case of T-number=30:
1 5(30)-63+5(1)=87+5
0
2 5(30)-63+5(2)=87+10
0
0 5(30)-63+45(1)=87+45
1
0 5(30)-63+45(2)=87+90
2
Below we give an example of the application of the rule when T-shape moves horizontally and vertically (right across and downwards) as follows:
If T-number=30 and assuming that the T=shape moves according to the below translation vector::
2 5(59)-63+5(2)+45(3)=232+10+135
3
The general rule for the two-dimension movement of the T-shape is the following:
5n-63+5(x)+45(y)
