-As we can see, when T-Number increases by 1, T-Total also increases by 5. It is obvious that there is a certain pattern.
-If I define T-Number = N
T-Total = N+N-10+N-20+N-21+N-19 = 5N-70 = 5N-70
-To check whether it is right, it is easy and quick that I substitute N for 22.
→5N-70 = 5*22-70 = 110-70 = 40
-When T-Number is 22, T-total is 40 and equation fits. Consequently the relationship between T-Number and T-Total is :
→T = 5N-70
-Now I use 11 by 11 number grid to find the relationship between T-Number and T-Total.
Sample 1)
T-Number = 24
T-Total = 1+2+3+13+24 = 43
Sample 2)
T-Number = 25
T-Total = 2+3+4+14+25 = 48
Sample 3)
T-Number = 26
T-Total = 3+4+5+15+26 = 53
-I arrange the samples in the table.
-Again we notice, when T-Number increases by 1, T-Total also increases by 5. Now it is certain that there is a pattern.
-If I define T-Number = N
T-Total = N+N-11+N-22+N-23+N-21 = 5N-77
-Now I am going to check whether it it true. To do it, I substitute N for 24.
→5N-77 = 5*24-77 = 120-77 = 43
-When T-Number is 24, T-Total is 43 and the equation does make 43 when N is 24. Therefore the relationship between T-Number and T-Total is :
→T = 5N-77
-Now, I arrange these 3 equations that I have found from my the investigation.
-Look at the numbers -63, -70, and -77. They increase by -7 when the size of grid increases by 1.
-From these numbers, we can find a certain interesting point.
→ -63 = -7*9
→ -70 = -7*10
→ -77 = -7*11
-When we look at these results, we can notice that - 7 is constant and the number which we use to multiply it is the same as grid number. For example, -63 is from the equation of 9 by 9 grid and - 63 = -7*9. The same relationship is found in all the examples and therefore the relationship between T-Number, T-Total, and number grid is :
→T-Number = N, T-Total = T, Grid =G
T = 5N-7G
-Now, I turn T-shape 90 degree clockwise and investigate the relationship between T-Number and T-Total.
Original Rotated one
* = T-Number
-I use 9 by 9 grid to find the relationship.
Sample 1)
T-Number = 10
T-Total = 10+11+12+3+21 = 57
Sample 2)
T-Number = 11
T-Total = 11+12+13+4+22 = 62
Sample 3)
T-Number = 12
T-Total = 12+13+14+5+23 = 67
-I arrange these samples in the table.
-It shows that there is a certain pattern as when T-Number increases by 1, T-Total increases by 5.
-If I define T-Number = N
T-Total = N+N+1+N+2+N+11+N-7 = 5N+7
-I check whether it is a suitable equation, by substituting N for 10
→5N+7 = 5*10+7 = 57
-As I checked above, it is the right equation. Therefore the relationship between T-Number and T-Total is :
→T = 5N+7
-I Change the grid to 10 by 10 and investigate the relationship.
Sample 1)
T-Number = 11
T-Total = 11+12+13+3+23 = 62
Sample 2)
T-Number = 12
T-Total = 12+13+14+4+24 = 67
Sample 3)
T-Number = 13
T-Total = 13+14+15+5+25 = 72
-I arrange the results in the table.
-As we notice, when T-Number increases by 1, T-Total increases by 5. It is certain that there is a pattern.
-If I define T-Number = N
T-Total = N+N+1+N+2+N-8+N+12 = 5N+7
-To check whether it is right, I substitute N for 11.
→5N+7 = 5*11+7 = 62
-This equation does make 62 when T-number is 11. As a result, the relationship between T-Number and T-Total is :
→T = 5N+7
-This time, I use 11 by 11 number grid to find the relationship.
Sample 1)
T-Number = 12
T-Total = 12+13+14+3+25 = 67
Sample 2)
T-Number = 13
T-Total = 13+14+15+4+26 = 72
Sample 3)
T-Number = 14
T-Total = 14+15+16+5+27 = 77
-I arrange the these results in the table.
-It is possible that there is a certain pattern as when T-Number increases by 1, T-Total increases by 5.
-If I define T-Number = N
T-Total = N+N+1+N+2+N-9+N+13 = 5N+7
-I have to check this equation to prove that it is the right equation.
→5N+7 = 5*12+7 = 67
-The result above show that this equation is the right one. Therefore the relationship between T-Number and T-Total is :
→T = 5N+7
-Using the equations I found, I make a table.
-When we look at the table, we can find a interesting point. The point is that the equation is constant although the grid is getting larger. Therefore the relationship between T-Number, T-Total, and Grid is :
→Whatever the grid is, the relationship between T-Number and T-Total in T = 5N+7.
-I turn the original T-shape 180 degree clockwise and investigate the relationship between T-Number and T-Total.
Original Rotated one
* = T-Number
-First, I use 9 by 9 number grid.
Sample 1)
T-Number = 2
T-Total = 2+11+20+19+21 = 73
Sample 2)
T-Number = 3
T-Total = 3+12+21+20+22 = 78
Sample 3)
T-Number = 4
T-Total = 4+13+22+21+23 = 83
-I arrange these samples in the table.
-As we can notice, when T-Number increases by 1, T-Total increases by 5. It shows that there is a certain pattern.
-If I define T-Number = N
T-Total = N+N+9+N+18+N+17+N+19 = 5N+63
-To prove it is the right equation, I have to check this equation by substituting N for 2.
→5N+63 = 5*2+63 = 73
-The result above shows that this equation is the right one. Therefore the relationship between T-Number and T-Total is :
→T = 5N+63
-I change the grid to 10 by 10 and investigate the relationship.
Sample 1)
T-Number = 2
T-Total = 2+12+22+21+23 = 80
Sample 2)
T-Number = 3
T-Total = 3+13+23+22+24 = 85
Sample 3)
T-Number = 4
T-Total = 4+14+24+23+25 = 90
-I arrange these results in the table.
-When T-Number increases by 1, T-Total increases by 5. It is possible that there is a certain pattern.
-If I define T-Number = N
T-Total = N+N+10+N+20+N+19+N+21 = 5N+70
-To check whether it is right, I substitute N for 2.
→5N+70 = 5*2+70 = 80
-This equation does make 80 when N is 2. Therefore the relationship between T-Number and T-Total is :
→T = 5N+70
-This time, I use 11 by 11 grid to find the relationship between T-Number and T-Total.
Sample 1)
T-Number = 2
T-Total = 2+13+24+23+25 = 87
Sample 2)
T-Number = 3
T-Total = 3+14+25+24+26 = 92
Sample 3)
T-Number = 4
T-Total = 4+15+26+25+27 = 97
-I arrange these results in the table.
-As we can notice, it is possible that there is a certain pattern because when T-Number increases by 1, T-Total increases by 5.
-If I define T-Number = N
T-Total = N+N+11+N+22+N+21+N+23 = 5N+77
-To check whether it is the right equation, I substitute N for 2.
→5N+77 = 5*2+77 = 87
-The result above shows that this equation does make 87 when N is 2 so that it is the right equation. Therefore the relationshio between T-Number and T-Total is :
→T = 5N+77
-Now, I arrange these 3 equations that I have found from the investigation.
-Look at the numbers 63, 70, 77. They increase by 7 when the size of grid increases by 1.
-From these numbers, we can find an interesting point.
→63 = 7*9
→70 = 7*10
→77 = 7*11
-When we look at these results, we can notice that 7 is constant and the number which we use to multiply it is the same as the grid number. For example, 63 is from the equation of the 9 by 9 grid and 63 = 7*9. The same relationship is found in all the examples and therefore the relationship between T-Number, T-Total, and number grid is :
→T-Number = N, T-Total = T, Grid = G
T = 5N+7G
-What I have found from the investigation is that T-Total increases by a certain algebraic pattern linked with T-Number and Grid number and algebraic pattern is changed when number grid is changed and/or T-shape is rotated. Therefore the relationship between T-Number and T-Total is variable.
