N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63
Horizontal translation
My T number is 24 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 24+15+6+5+7= 57. The number in my T shape directly above my T number is 9 places back on the grid so it is N-9.The number in my T shape directly above this is also 9 places back on the grid so it is N-9-9= N-18. The two remaining numbers in the T shape are N-18+1 and N-18-1. Thus the T total is:
N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63
Horizontal translation
My T number is 25 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 25+16+7+6+8= 62. The number in my T shape directly above my T number is 9 places back on my grid so it is N-9.The number in my T shape directly above this is also 9 places back on the grid so it is N-9-9= N-18. The two remaining numbers in the T shape are N-18+1 and N-18-1. Thus the T total is:
N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63
T-Number T-Total
20 37
21 42
22 47
23 52
24 57
25 62
Looking at my results above I have spotted a relationship between the T-Totals as I transform the T-shape horizontally. I have noticed that as I transform the T-shape horizontally my T-Total is increased by 5 I will test my theory by making a prediction on my T-Number 26. My prediction is that when my T-Number is 26 my T-Total will be 67.
Horizontal translation
My T number is 26 as you can see on my 9by9 grid and I will also be representing this as N in my equation. My T total is 26+17+8+7+9= 67. The number in my T shape directly above my T number is 9 places back on the grid so it is N-9. The number directly above this is also 9 places back on my grid so it is N-9-9= N-18. The remaining two numbers are N-18+1 and N-18-1. Thus the T total is:
N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = 5N-63
I have shown above that my prediction was correct.
Un 1 2 3 4 5 7 8
37 42 47 52 57 62 67
Diff 5 5 5 5 5 5
Un= n+5
My general expression for my 9by9 grid is:
N-19 N-18 N-17
N-9
N
N+ (N-9) + (N-18) + (N-18+1) + (N-18-1) = N
My T-Number is 18 which you can see in my 8by8 grid above and I will be representing this as N in my equation. My T-total is 18+10+2+1+3 = 34. The number in my T-shape directly above my T-number is 8 places back on my grid so it is N-8. The number directly above this is also 8 places back on the grid so it is N-8-8 = N-16. The remaining two numbers are N-16-1 and N-16+1. Thus the T-total is:
N+ (N-8) + (N-16) + (N-16+1) + (N-16-1) = 5N-52
My T-Number is 19 which you can see in my 8by8 grid above and I will be representing this as N in my equation. My T-total is 19+11+3+2+4= 39. The number in my T-shape directly above my T-number is 8 places back on my grid so it is N-8. The number directly above this is also 8 places back on the grid so it is N-8-8= N-16. The remaining two numbers are N-16-1 and N-16+1. Thus the T-total is:
N+ (N-8) + (N-16) + (N-16+1) + (N-16-1) = 5N-52
My T-Number is 20 which you can see in my 8by8 grid above, and I will be representing this as N in my equation. My T-total is 20+12+4+3+5= 44. The number in my T-shape directly above my T-number is 8 places back on my grid so it is N-8. The number directly above this is also 8 places back on my grid so it is N-8-8= N-16. The two remaining numbers in my grid are N-16-1 and N-16+1. Thus my T-total is:
N+ (N-8) + (N-16) + (N-16-1) + (N-16+1) = 5N-52
My T-Number is 21 which you can see in my 8by8 grid above, and I will be representing this as N in my equation. My T-total is 21+13+5+4+6= 49. The number in my T-shape directly above my T-number is 8 places back on my grid so it is N-8. The number directly above this is also 8 places back on my grid so it is N-8-8= N-16. The two remaining numbers in my grid are N-16-1 and N-16+1. Thus my T-total is:
N+ (N-8) + (N-16) + (N-16-1) + (N-16+1) = 5N-52
My T-Number 22 which you can see in my 8by8 grid above and I will be representing as N in my equation. My T-total is 22+14+6+5+7= 54. The number in my T-shape directly above my T-number is 8 places back on my grid so it is N-8. The number directly above this is also 8 places back on the grid so it is N-8-8= N-16. The remaining two numbers are N-16-1 and N-16+1. Thus the T-total is:
N+ (N-8) + (N-16) + (N-16-1) + (N-16+1) = 5N-52
T-number T-total
18 34
19 39
20 44
21 49
22 54
Analyzing the results above I have noticed a relationship between the T-total and the T-number. The relationship I have discovered is that as my T-number increases by 1 my T-total increases by 5 I will then go on to try and prove this theory by predicting the 23rd T-number. I predict my 23rd T-numbers T-total been 59.
My T-number is 23 as you can see from my 8by8 grid above I will be representing this as N in my equation. My T-total is 23+15+7+6+8= 59. The number in my T-shape directly above my T-number is 8 places back on my grid so it is represented as N-8. The number directly above this is also 8 places back on my grid so it is represented as N-8-8= N-16. The remaining two numbers are N-16-1 and N-16+1. Thus my T-total is:
N+ (N-8) + (N-16) + (N-16-1) + (N-16+1) = 5N-52
Un 1 2 3 4 5 6
My T-number is 16 as you can see above on my 7by7 grid I will be representing this as N in my equation. My T-total is 16+9+2+1+3= 31. The number in my T-shape directly above my T-number is 7 places back on my grid so it is represented as N-7. The number directly above this is also 7 places back on my grid so this is represented as N-7-7= N-14. The two numbers remaining in my T-shape are N-14-1 and N-14+1. Thus my T-total is:
N+ (N-7) + (N-14) + (N-14-1) + (N-14+1) = 5N-49
My T-number is 17 as you can see above on my 7by7 grid I will be representing this as N in my equation. My T-total is 17+10+3+2+4= 36. The number in my T-shape directly above my T-number is 7 places back on my grid so it is represented as N-7. The number directly above this is also 7 places back on my grid so it is represented as N-7-7= N-14. The two remaining numbers in my T-shape are N-14-1 and N-14+1. Thus my T-total is:
N+ (N-7) + (N-14) + (N-14-1) + (N-14+1)= 5N-49
My T-number is 18 as you can see above on my 7by7 grid I will be representing this as N in my equation. My T-total is 18+11+4+3+5= 41. The number in my T-shape directly above my T-number is 7 places back on my grid so it is represented as N-7. The number directly above this is also 7 places back on my grid so it is represented as N-7-7= N-14. The two numbers remaining in my T-shape are N-14-1 and N-14+1. Thus my T-total is:
N+ (N-7) + (N-14) + (N-14-1) + (N-14+1) = 5N-49
My T-number is 19 as you can see above on my 7by7 grid I will be representing this as N in my equation. My T-total is 19+12+5+4+6= 46. The number in my T-shape directly above my T-number is 7 places back on my grid so it is represented as N-7. The number directly above this is also 7 places back on my grid so it is represented as N-7-7= N-14. The two remaining numbers in my T-shape are N-14-1 and N-14+1. Thus my T-total is:
N+ (N-7) + (N-14) + (N-14-1) + (N-14+1) = 5N-49
T-total T-number
16 31
17 36
18 41
19 46
Using this data in my table I will go on to try and predict the 20th T-number in my 7by7 grid. Analyzing my data I have picked up on a relationship that as my T-number increases by 1 my T-total increases by 5.
My T-number is 20 as you can see above on my 7by7 grid I will be representing this as N in my equation. My T-total is 20+13+6+5+7= 51. The number in my T-shape directly above my T-number is 7 places back on my grid so it is represented as N-7. The number directly above this is also 7 places back on my grid so it is represented as N-7-7= N-14. The two remaining numbers in my T-shape are N-14-1 and N-14+1. Thus my T-total is:
N+ (N-7) + (N-14) + (N-14-1) + (N-14+1) = 5N-49
I have proven my prediction right that my T-number is 51 which is a increase of 5 from my previous T-total 46.
My T-number is 14 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 14+8+2+1+3= 28. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid so therefore it is represented as N-6-6= N-12. The two remaining numbers in my T-shape are N-12-1 and N-12+1. Thus my T-total is:
N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42
My T-number is 15 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 15+9+3+2+4= 33. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid therefore it is represented as N-6-6= N-12. The two remaining numbers in my T-shape are N-12-1 and N-12+1. Thus my T-total is:
N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42
My T-number is 16 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 16+10+4+3+5= 38. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid so it is represented as N-6-6= N-12. The two remaining numbers are N-12-1 and N-12+1. Thus my T-total is:
N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42
T-number T-total
14 28
15 33
16 38
Using this data in my table I will try and predict my 17th T-number in my 6by6 grid. Analyzing the data in my table I have picked up on a sequence which is when my T-number increases by 1 my T-total increases by 5.
My T-number is 17 as you can see above on my 6by6 grid I will be representing this as N in my equation. My T-total is 17+11+5+4+6= 43. The number in my T-shape directly above my T-number is 6 places back on my grid so therefore it is represented as N-6. The number directly above this is also 6 places back on my grid so it is represented as N-6-6= N-12. The two remaining numbers in the T-shape are N-12-1 and N-12+1. Thus my T-total is:
N+ (N-6) + (N-12) + (N-12-1) + (N-12+1) = 5N-42
I have proven my prediction correct because my T-total is 43 which is an increase from the 16th T-number which was 38.
