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 T 69 = 50+51+52+60+69 = 282 T 22 = 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-19 N-18 N-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. Ill put that idea into another T. Note W = width number (9) N- (2 W-1) N-2 WN- (2 W+1) N-W N This i the same thing as before but shown algebraically. The formula for the Value of the T-total now is shown as: 5 N-7 W = T-total Aim 2- different sizes and relationship I know this works for the grid 9 by 9 but Im not sure if itll 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 T 22 = 1+2+3+12+22 = 42 I notice this is 5 more than 9 by 9 T 69 = 48+49+50+59+69 = 275 Obviously 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 its 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 = 7 W 12 by 12 123456789101112 131415161718192021222324 252627282930313233343536 373839404142434445464748 495051525354555657585960 616263646566676869707172 737475767778798081828384 858687888990919293949596 979899100101102103104105106107108 109110111112113114115116117118119120 121122123124125126127128129130131132 133134135136137138139140141142143144 T 62 = 37+38+39+50+62 also = (625) - (712) = 226 = 226 T 141 = 116+117+118+129+141 also = (1415) - (712) = 621 = 621 Aim 3- Transformations stretches and there effects on the formula Ill 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 ST 64 as it starts at 64, its a stretch of 2 in both directions.

St 64 = 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: 8 N-18 W = T-total 8 N = number of integers in the T-shape 18 W = 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.