So I think the number four might also be important in this investigation.
4+5=9 and my square is a nine by nine square so this could be important.
Because I thought five was going to be important in this investigation I will multiply the T-number by five and then take away the T-total to see if this shows a pattern
It was a nine by nine grid and 63 is in the 9 time tables it equals 7x9 so seven might also be important in this investigation.
Here I have made a formula to see if this works on other grid sizes.
T-number = n
5n – T-total = 7x9 (grid size multiplied by seven)
I will now try this on other grid sizes starting with a 5by5 grid.
T- total 1+2+3+7+12=25
T- Number= 13
T- total=2+3+4+8+13= 30
T- Number= 13
T-total= 3+4+5+9+14= 35
T- Number= 14
From this I can see that every time the T- number goes up by one and the t- total goes up by five.
I think this is because there are five numbers in the T- shape.
Now I will look at the relationship between the T- total and the T- number.
From this I can see a pattern that each time the T- number goes up by one. The difference between the T- numbers to the T- total goes up by four.
Because five is an important number in this investigation I will multiply the T- number by 5 times table then take away the T- total to see if this shows a pattern.
It was a five by five grid and 35 is in the five times tables is it is 7x5. So seven is important in this investigation because before I found that the difference was 63 and that was 7x9.
From this I can see my formula will work
5n – T-total = 7x5 (grid size multiplied by seven)
5x12– 60=35
5x13- 60=35
5x14- 60=35
Now I will try a six by six grid to see if my formula works.
T-total= 1+2+3+8+14=28
T-number= 14
T-total= 2+3+4+9+15=33
T-number=15
T-total= 3+4+5+10+16=38
T-number=16
T-total= 4+5+6+11+17=43
T-number=17
Here is a table to show the T-total and the T-number of this six by six grid.
I will now find the difference between the T-total and the T-number.
From this I can see that each time the T-number goes up by one the difference goes up by four.
Because five is an important number in this investigation I will multiply the T-number by five then take away the T-total and see if I can see a pattern.
I will try my formula on each of these T-numbers and multiply them by five then minus 63 to see if I get the T-total for each one.
5n-63
5x20-63=37
5x21-63=42
5x22-63=47
5x23-63=52
5x24-63=57
I will now predict what I will get for 25,26 and 27.
For 25 I think the answer will be 62
For 27 I think the answer will be 67
For 28 I think the answer will be 72
I will now try my formula and see if I am correct.
5n-63
5x25-63=62 The T-total for 25= 6+7+8+16+25 = 62
5x26-63=67 The T-total for 26= 7+8+9+17+26 = 67
5x27-63=72 The T-total for 27= 8+9+10+18+27=72
This proves that my formula does work.
I will now try and see if this formula works on the T-numbers on a five by five grid. Here is a table to show the T-totals and the T-numbers.
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I am going to use the Formula 5n-35. I will use 5 because there is five numbers in the T-shape and I will minus 35 because that was the difference between the T-total and the T-number each time.
Here is a table to show how I found the difference between the T-number and the T-total. I did this by multiplying the T-number by five each time then I would I minus the T-total and I would get the same answer for the difference each time.
I will try my formula on each of these T-numbers and multiply them by five then minus 35 to see if I get the T-total for each one.
5n-35
5x12-35=25
5x13-35=30
5x14-35=35
I will now predict what I will get for 15,16 and 17
For 17 I think I will get 50
For 18 I think I will get 55
I will now try my formula to see if I am correct
5x17-35=50 The T-total for 17= 6+7+8+12+17=50
5x18-35=55 The T-total for 18=7+8+9+13+18=55
This proves that my formula does work.
For both of these grids I have found that the number you minus is the grid number multiplied by 7 so my formula is 5n-the grid number multiplied by seven. So for a…
5by5grid the formula will be 5n-35
6by6 grid the formula will be 5n-42
7by7 grid the formula will be 5n-49
8by8 grid the formula will be 5n-56
9by9 grid the formula will be 5n-63
I will now try and find a Formula to find the T-total but for T-shapes in different positions.
I will start with side down T-shapes On a 9by9 grid.
T-total= 2+11+19+20+21=73
T-number=2
T-total=3+12+20+21+22=78
T-number=3
T-total=4+13+21+22+23=83
T-number=4
T-total=5+14+22+23+23=88
T-number=5
T-total=6+15+23+24+25=93
T-number=6
I will now try a formula I will use the formula I used on the other 9by9 grid which was 5n-63
5x2-63=-53
This will not work so I will try adding instead of misusing
5n+63
5x2+63=73, which was our T-total
T-total= 2+11+19+20+21=73. I will now check this on the other T-totals I did.
5n+63
5x3+63=78 which was the T-total 3+12+20+21+22=78
5x4+63=83 which was the T-total 4+13+21+22+23=83
5x4+63=88 which was the T-total 5+14+22+23+23=88
5x5+63+93 which was the T-total 6+15+23+24+25=93
To make sure my formula works I will try this on a 7by7 grid. I predict my formula will be 5n+49
T-total=2+9+16+15+17=59
T-number=2
T-total=3+10+16+17+18=64
T-number=3
T-total=4+11+17+18+19=69
T-number=4
T-total=5+12+18+19+20=74
T-number=5
T-total=6+13+19+20+21=79
T-number=6
I will now try the formula I predicted which was 5n+49.
5x2+49=59 which was the T-total 2+9+16+15+17=59
5x3+49=64 which was the T-total 3+10+16+17+18=64
5x4+49=69 which was the T-total 4+11+17+18+19=69
5x5+49=74 which was the T-total 5+12+18+19+20=74
5x6+49=79 which was the T-total 6+13+19+20+21=79
For both of these grids I have found that the number you add is the grid number multiplied by 7 so my formula is 5n+the grid number multiplied by seven. So for a…
5by5 grid the formula will be 5n+35
6by6 grid the formula will be 5n+42
7by7 grid the formula will be 5n+49
8by8 grid the formula will be 5n+56
9by9 grid the formula will be 5n+63
I will now try and find a formula for the T-shape on its side.
5by5grid the formula will be 5n+35
6by6 grid the formula will be 5n+42
7by7 grid the formula will be 5n+9
8by8 grid the formula will be 5n+6
9by9 grid the formula will be 5n+3
