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.