The random number I have chosen is 69; this will be the t-number for checking my formula. I am now going to substitute the number into the equation:
T-total = 5(69)-63
T-total = 282
Now that I have got my prediction I will do what I have done for all the other numbers and find it out manually.
Predicted number:
The t-number in this case will be 69.
The t-total is 69+60+51+50+52 which will
give us 282.
T-number: 69
T-total: 282
My formula has turned out to be correct, I will keep this information and move on to the second grid.
Grid 2: (10 by 10)
Secondly I have chosen to look at the 10 by 10 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five and then I will be moving onto the third grid.
Number 1:
The t-number in this case will be 44.
The t-total is 44+34+24+23+25 which will give us 150.
T-number: 44
T-total: 151
Number 2:
The t-number in this case will be 45.
The t-total is 45+35+25+24+26 which will
give us 155.
T-number: 45
T-total: 155
Number 3:
The t-number in this case will be 46.
The t-total is 46+36+26+25+27 which will
give us 160.
T-number: 46
T-total: 160
Number 4:
The t-number in this case will be 47.
The t-total is 47+37+27+26+28 which will
give us 165.
T-number: 47
T-total: 165
Number 5:
The t-number in this case will be 48.
The t-total is 48+38+28+27+29 which will
give us 170.
T-number: 48
T-total: 170
Formula:
After investigating the t-numbers from 44 to 48 and comparing them with their t-totals, I have noticed that every time I increase the t-number by one the t-total goes up by five. I found this out by doing the table seen below:
This table shows us that 5 multiplied by the nth term added to 145 will give us the t-total.
Overall formula:
I have also figured out a formula that will give us the t-total of any chosen t-number at random on a 10 by 10 grid. I did this by calling the t-number the nth term in each case and working out a formula from there, what I have done can be seen looking at the table below:
*nth term
Formula: 5n-70.
Checking:
I will now be checking if the formula I have come up with was correct, I will do this by choosing a random number on the 10 by 10 grid and substituting the t-number into the equation and coming up with the t-total without adding up the rest of the numbers. I will then add up the t-total and if it matches my prediction my formula will be correct.
The random number I have chosen is 72; this will be the t-number for checking my formula. I am now going to substitute the number into the equation:
T-total = 5(72)-70
T-total = 290
Now that I have got my prediction I will do what I have done for all the other numbers and find it out manually.
Predicted number:
The t-number in this case will be 72.
The t-total is 72+62+52+51+53 which will
give us 290.
T-number: 72
T-total: 290
My formula has turned out to be correct, I will keep this information and move on to the third grid.
Grid 3: (11 by 11)
Thirdly I have chosen to look at the 11 by 11 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five and then I will be moving onto the final grid.
Number 1:
The t-number in this case will be 83.
The t-total is 83+72+61+62+60 which will give us 338.
T-number: 83
T-total: 338
Number 2:
The t-number in this case will be 84.
The t-total is 84+73+62+63+61 which will
give us 343.
T-number: 84
T-total: 343
Number 3:
The t-number in this case will be 85.
The t-total is 85+74+63+64+62 which will
give us 348.
T-number: 85
T-total: 348
Number 4:
The t-number in this case will be 86.
The t-total is 86+75+64+65+63 which will
give us 353.
T-number: 86
T-total: 353
Number 5:
The t-number in this case will be 87.
The t-total is 87+76+65+66+64 which will
give us 358.
T-number: 87
T-total: 358
Formula:
After investigating the t-numbers from 83 to 87 and comparing them with their t-totals, I have noticed that every time I increase the t-number by one the t-total goes up by five. I found this out by doing the table seen below:
This table shows us that 5 multiplied by the nth term added to 333 will give us the t-total.
Overall formula:
I have also figured out a formula that will give us the t-total of any chosen t-number at random on a 11 by 11 grid. I did this by calling the t-number the nth term in each case and working out a formula from there, what I have done can be seen looking at the table below:
*nth term
Formula: 5n-77.
Checking:
I will now be checking if the formula I have come up with was correct, I will do this by choosing a random number on the 11 by 11 grid and substituting the t-number into the equation and coming up with the t-total without adding up the rest of the numbers. I will then add up the t-total and if it matches my prediction my formula will be correct.
The random number I have chosen is 105; this will be the t-number for checking my formula. I am now going to substitute the number into the equation:
T-total = 5(105)-77
T-total = 448
Now that I have got my prediction I will do what I have done for all the other numbers and find it out manually.
Predicted number:
The t-number in this case will be 105.
The t-total is 105+94+83+84+82 which will
give us 448.
T-number: 105
T-total: 448
My formula has turned out to be correct; I will keep this information and move on to the final grid.
Grid 4: (12 by 12)
Lastly I will be looking at the 12 by 12 grid. I will be taking five t-numbers in a row and investigating the t-totals for them. Once I have completed all five, I will then look for a formula to link those five and then I will be moving onto the final grid.
Number 1:
The t-number in this case will be 50.
The t-total is 50+38+25+26+27 which will give us 166.
T-number: 50
T-total: 166
Number 2:
The t-number in this case will be 51.
The t-total is 51+39+27+26+28 which will
give us 171.
T-number: 51
T-total: 171
Number 3:
The t-number in this case will be 52.
The t-total is 85+40+28+27+29 which will
give us 176.
T-number: 52
T-total: 176
Number 4:
The t-number in this case will be 53.
The t-total is 53+41+29+30+28 which will
give us 181.
T-number: 53
T-total: 181
Number 5:
The t-number in this case will be 54.
The t-total is 54+42+30+31+29 which will
give us 186.
T-number: 54
T-total: 186
Formula:
After investigating the t-numbers from 50 to 54 and comparing them with their t-totals, I have noticed that every time I increase the t-number by one the t-total goes up by five. I found this out by doing the table seen below:
This table shows us that 5 multiplied by the nth term added to 161 will give us the t-total.
Overall formula:
I have also figured out a formula that will give us the t-total of any chosen t-number at random on a 12 by 12 grid. I did this by calling the t-number the nth term in each case and working out a formula from there, what I have done can be seen looking at the table below:
*nth term
Formula: 5n-84.
Checking:
I will now be checking if the formula I have come up with was correct, I will do this by choosing a random number on the 12 by 12 grid and substituting the t-number into the equation and coming up with the t-total without adding up the rest of the numbers. I will then add up the t-total and if it matches my prediction my formula will be correct.
The random number I have chosen is 130; this will be the t-number for checking my formula. I am now going to substitute the number into the equation:
T-total = 5(130)-84
T-total = 566
Now that I have got my prediction I will do what I have done for all the other numbers and find it out manually.
Predicted number:
The t-number in this case will be 130.
The t-total is 105+106+107+118+130 which will give us 566.
T-number: 130
T-total: 566
My formula has turned out to be correct.
Conclusion:
After carefully examining each formula carefully I have come up with a formula that will link any t-number on any grid to its t-total. I firstly collected all the formulas for the grids and put them in this table:
The first thing I noticed when I looked at this table that every time the grids size was increased by one, the number taken away from 5n in the formula increases by seven.
I also noticed that the number taken away from 5n is the same as seven multiplied by the grid number every time, so 9 would be 63, 15 would be 105 and so on.
By figuring this out I managed to make a formula linking the t-number of any grid to its t-total using a quick simple formula. The formula is as follows:
T-total = 5n-7g
N=the t-number
G= the grid number
Now I will simply try the formula out on a random t-number chosen on a random grid and see if my formula works out correctly.
The random t-number I have chosen is 109 from the grid 11 by 11. I am now going to substitute these figures into the equations I have got and predict the t-total without adding the numbers. I will then check my prediction and if it matches the answer I add up I will know that my formula is correct.
Prediction:
T-total = 5n-7g (n=109 , g=11)
(5*109) - (7*11 545-77
My prediction is that the t-total for t-number 109 is going to be 468. I will now check to see if my answer is correct.
Check formula:
The t-number in this case will be 109.
The t-total is 109+98+87+86+88 which will
give us 468.
T-number: 109
T-total: 468
My formula is correct; I have solved the relationship between t-numbers and t-totals on any sized grid. The formula (5n-7g) is the answer to this problem.
