T- Total Essay, Research Paper
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Look
at the T-shape drawn on the 9 by 9 number grid. The total of the numbers
inside the T-shape is 1+2+3+11+20=37 This is called the
T-total. The number at the bottom
of the T-shape is called theT-number. The T-number for this
T-shape is 20. T ????? If
you take the other numbers in the T-Shape away from the T-Number you get a
T-Shape like this. T-17 T-18 T-19 T-9 T You
will notice that the centre column of the T-Shape is going up in 9?s because of
the table size. With the table set out like this a formula can be worked out to
find any T-Total on this size grid. This is done in the working below:- T-total?????? = ???? T-19+T-18+T-17+T-9+T ???????? ?????? =????? 5T-63 Now to test this formula
to see if it works For T-Total I will use the
letter X For
the T-Number I will use the letter TSo X = 5T-63 ???? T = 20 ???? X = 5×20-63 ???????? = 100-63 ???????? = 37 Now
I will do two more to check to see if it will work anywhere on the grid.T=43 X=5×43-63???????????????????????? 24+25+26+34+43=152 X=215-63 X=152T=49 X=5×49-63???????????????????????? 30+31+32+40+49=182 X=245-63 X=182I
have tested the formula for three different T-numbers and the formula works for
a 9 by 9 grid. The
full formula for this size grid is:- X=5T-63 ??????????? When
trying to fine out the T-Total in a 6 by 6 number grid, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 If
you take the other numbers in the T-Shape away from the T-Number you get a
T-Shape like this. T-13 T-12 T-11 T-6 T This
time the centre column of the T-Shape is going up in 6?s because of the table
size. The formula for this size grid is:- T-Total????? = T-13+T-12+T-11+T-6+T ?????????? ??? ?= 5T-42Now to test this formula
to see if it works, again For the T-total I will use
the letter X For
the T-Number I will use the letter TSo X = 5T-42 ???? T = 14 ???? X = 5×14-42 ???????? = 70-42 ???????? = 28Now
I will do two more to check to see if it will work anywhere on the grid.T=22 X=5×22-42???????????????????????? 9+10+11+16+22=68 ? =110-42 ? = 68 T=35 X=5×35-42???????????????????????? 22+23+24+29+35=133 ? =175-42 ? =133I
have tested the formula for three different T-numbers and the formula works for
a 6 by 6 grid. The
full formula for this size grid is:- X=5T-42Below
is a grid with parts of the formula in. I will use this to figure out a formula for
any T-Total on any width grid. T-2G-1 T-2G T-2G+1 T-G T T-total = T-2G-1 ????????????? T-2G ????????????? T-2G+1 ????????????? T-G ????????????? T X=5T-7GIf the Grid Width (= G),
and the T-Number (= T), the T-Total (= X) is calculated by ??????????? X=5T-7G
Now I?m going to try the
grid that I drew earlier i.e. G=9 The formula is????? X=5T-7G Which changes to X=5T-7×9 ??????? ?????????????????
X=5T-63I
will now try the formula on a 7 by 7 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 X=5X17-49 (2+3+4+10+17=36 X=85-49 X=36I
am now going to try this formula on a 3 by 5 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X=5X8-21 (1+2+3+5+8=19 X=40-21 X=19I am now going to try this
formula on a 6 by 10 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 X=5X33-42
(20+21+22+27+33=123 X=165-42 X=123 Now I have worked the formula out for the T-Shape
in the one position, I am going to translate the T-Shape to different positions
and investigate the relationships between the T-Total, the T-number and the
grid size.The other T-Shape?s that I can find a formula?s are
shown below:- ???????????????? A 3 by 3 grid can be used
for the other 3. 1 2 3 4 5 6 7 8 9 T T+G T+2G+1 T+2G T+2G-1 T T+G T+2G T+2G-1 T+2G+1 7G+5T X=7G+5T=7×3+5×2 ??????????????? =21+10 (2+5+7+8+9=31) ??????????????? =31I will now try this new formula on a 4 by 7 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 7G+5T x=7×4+5×10
(17+18+19+14+10=78) ? =28+50 ? =78 I will now try it on a 8
by 3 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 7G+5T x=7×9+5×2 (19+20+21+11+2=73) ? =56+10 ? =73 I have tried this new formula for an inverted T and it
works, the formula is: 7G+5T 1 2 3 4 5 6 7 8 9 T-1-G T-2 T-1 T T-2+G T T-1 T-2 T-2-G T-2+G 5T-7X= 5T-7=5×6-7 ??????????????? =30-7 (1+4+7+5+6=23) ??????????????? =23I
will now try the new formula on a 4 by 4 grid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5T-7 X=5×11-7 (5+9+13+10+11=48) ? =48And on a 4 by 9 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 5T-7 x=5×19-7 (13+17+21+18+19=88) ? =88 I have
tested this new formula for a side ways T the formula works it is: 5T-7
1 2 3 4 5 6 7 8 9 T+2-G T T+1 T+2 T+2+G T T+1 T+2 T+2+G T+2-G 5t+7 The formula is 5T+7 X=5T+7=5×4+7 ??????????????? =20+7 (3+6+9+5+4=27) ??????????????? =27 I will now try the new
formula on a 6 by 6 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ????????????????????????????? 5T+7 X=5×15+7 (11+17+23+16+15=82) ? =82 And on a 11 by 5 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 5T+7 X=5×28+7 (28+29+30+19+41=147) ? =147 I have
tested this new formula for a side ways T the formula works it is: 5T+7 The formula for a ?????? ?this way up is: X=5T-7G The formula for a ?????? ?this way up is: X=7G+5T The formula for a ?????? ?this way up is:X=5T+7 The formula for a ?????? ?this way up is: X=5T-7 ____________________________________________________ I
am now going to investigate the effect on the T-Total when the T-Shape is
transformed, rotated 90°, both
T-Shapes will have the same T-Number like this: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 T-
Number = 12 Purple
T-Total = 25 Grey T-Total = 67 a
difference of 42?.. The formula 5T ? 7G + 42 Now
I?m going to see what happens on a 6 by 6 width Grid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ? T-
Number = 14 Purple
T-Total = 33 Grey T-Total = 82 a
difference of 49?.. The formula 5T ? 7G + 497 by 7 width Grid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 T-
Number = 16 Purple
T-Total = 41 Grey T-Total = 97 a
difference of 56?.. The formula 5T ? 7G + 56Table of Results Grid width Purple T-Total Grey T-Total Difference between the two T-Total 5 25 67 42 6 33 82 49??????? Goes up in 7?s 7 41 97 56 _________________________________________________________________________________ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Now
I will rotate it again 180° from it?s original placeT-
Number = 12 Purple
T-Total = 25 Grey T-Total = 95? a
difference of 70?.. The formula 5T ? 7G + 70 Now
I?m going to see what happens on a 6 by 6 width Grid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 T-
Number = 14 Purple
T = 28 Grey T = 112????? a difference of 84?.. The formula 5T ?
7G + 84 7
by 7 Grid 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 T-
Number = 16 Purple
T-Total = 31 Grey T-Total = 129?? a difference of 98?.The formula 5T ? 7G
+ 98 Table of Results Now I will rotate it again
270&de