T-shapes Essay, Research Paper
Looking at the 9-9 grid below and the T-shape drawn on it,
The total number of the numbers on the inside of the T-shape is called the T-total
123456789
101112131415161718
192021222324252627
282930313233343536
373839404142434445
464748495051525354
555657585960616263
646566676869707172
737475767778798081
828384858687888990
The t-total for this T-shape is:
1+2+3+11+20=37
So 37 = T-total
The number at the bottom is the T-number, So the T-number for this shape is 20
Aims:
1)Investigate the relationship between the T-total and the T-number
2)Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size.
3)Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.
Aim 1- the solution
123456789
101112131415161718
192021222324252627
282930313233343536
373839404142434445
464748495051525354
555657585960616263
646566676869707172
737475767778798081
828384858687888990
T69= 50+51+52+60+69
=282
T22=3+4+5+13+22
=47
In the diagram below it shows the difference between the T-number and the other numbers. First is the T-shape in question:
123
11
20
This is the T-shape and here is the Difference T-shape:
N-19N-18N-17
N-9
N
This shows the difference N= T-number
In the T on the previous page I have noticed that the first difference from N is 9 which is also the Width of the square.
I?ll put that idea into another T. Note W= width number(9)
N-(2W-1)N-2WN-(2W+1)
N-W
N
This is the same thing as before but shown algebraically.
The formula for the Value of the T-total now is shown as:
5N-7W=T-total
Aim 2- different sizes and relationship
I know this works for the grid 9 by 9 but I?m not sure if it?ll work for any other grids.
Here is a test for a 10 by 10 grid
12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
T22=1+2+3+12+22
=42 I notice this is 5 more than 9 by 9
T69=48+49+50+59+69
=275Obviously no pattern there.
Method test?
(695)-70=275 YES it worked
My method seems to have worked out as it is logical and fairly straight forward to explain.
12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
As there are 10 in each row it?s obvious that the row above will be 10 less than the row below. So 68 is 10 less than the T-number 78. If you calculate the whole T you realise that row 2 is 10 less than row 1 and row 3 is 20 less than row 1,but there are three relevant numbers in row 3 which are 19 less and 21 less than the T-number. These cancel out to form 20 each, So finally we get (110)+(610)=(710)
=70
Or =7W
12 by 12
123456789101112
131415161718192021222324
252627282930313233343536
373839404142434445464748
495051525354555657585960
616263646566676869707172
737475767778798081828384
858687888990919293949596
979899100101102103104105106107108
109110111112113114115116117118119120
121122123124125126127128129130131132
133134135136137138139140141142143144
T62=37+38+39+50+62 also =(625)-(712)
= 226 =226
T141=116+117+118+129+141 also =(1415)-(712)
=621 =621
Aim3- Transformations stretches and there effects on the formula
I?ll do this with a 12 by 12 first, as this will give me enough accuracy to start with.
123456789101112
131415161718192021222324
252627282930313233343536
373839404142434445464748
495051525354555657585960
616263646566676869707172
737475767778798081828384
858687888990919293949596
979899100101102103104105106107108
109110111112113114115116117118119120
121122123124125126127128129130131132
133134135136137138139140141142143144
Stretch A will be called ST64 as it starts at 64, it?s a stretch of 2 in both directions.
St64=26+27+28+29+30+40+52+64
=296
I think I can work out the formula using my previous method so:
12+24+36+(436)=216
21612=18
This means the formula is:
8N-18W=T-total
8N= number of integers in the T-shape
18W=difference number calculated
Conclusion:
The size of the T-shape calculates the number before N in the formula and the grid size calculates the value of W. the number before W is calculated by looking at the rows and finding how many rows away from the T-number they are. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.
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