If n = 30 then 5x30-63= 87 and that is correct as 30+21+12+11+13=87. I will now check my formula on a randomly placed T-shape.
N=74 5x74-63=307
74+65+56+55+57=307
I have now proved that my formula works on a 9by9 grid so now I am going to work out the formula for a 10by10 grid. I am going to do this the same way I did it for the 9by9 grid.
N=83 83x5-70=345
62+63+64+73+83=345
N=57 57x5-70=215
57+47+37+36+38=215
My formula does work anywhere on the grid so now I am going to work out the formula for 8 by 8 and see if I can find something that relates them all together.
This is the first T-shape on an 8by 8 grid:
Adding up all the Ns and the numbers I get: 5n-56. So if N is 18 then the formula becomes 18x5-56=34 and 1+2+3+10+18=34. I shall now test the formula on other points in the grid.
With this shape n is 58: 58x5-56=234 and 58+50+42+41+43=234. I shall try it with one more shape just to make sure.
N is 31 so the formula is: 31x5-56=99 and 31+23+14+15+16=99. I have proved now that the formula works for this 8by 8 grid. I am now going to put all of my results into a table and see if there is a link for them.
By looking at my results I can see that each time the grid size goes up by 1 the formula changes as the second number goes up by 7. And also the grid size (g) x7 =the second number. I predict that the formula will be 5n-7g. I am going to check it by working it out the same way as I did the other formulae by working out the T-shape in relation with N but this time I’m going to add G (grid size).
Adding up all of the Ns and the Gs and the numbers I get: 5n-7g. I am now going to test this formula on the T-shapes I used before. Firstly I will test the 8 by 8 grids.
N=18 and G=8 so the formula is 5x22-7x8=34 and 18+10+2+1+3=34
N=58 G=8. 5x58-7x8=234. 58+50+42+41+43=234.
N=31 g=8. 31x5-7x8=99 31+23+14+15+16=99
I have now proved that the formula works on an 8 by 8 grid now ill try it on 9 by 9.
9 by 9
N=74 5x74-9x7=307
74+65+56+55+57=307
5x20-7x9=37 20+11+2+3+1=37
5x30-7x9=87 11+12+13+21+30=87. I have proved my formula works on a 9 by 9 I now will try it on a 10 by 10.
10 by 10
. N=22 and G=10 so 5x22-7x10=40 and 22+12+2+1+3=40
N=83 G=10 83x5-7x10=345
62+63+64+73+83=345
N=57 G=10 57x5-7x10=215
57+47+37+36+38=215
I have now proved that my formula works on any sized grid anywhere on the grid.
Now I am going to work out the formula for the T-shape when it has been rotated by 90o. I shall go about it the same way as I did for all of the other grids. I will start on a 9 by 9 grid.
I think that the formula for any sized grid is 5n+7. I will test this on a 9 by 9 a 10 by 10 and an 8 by 8 grid. If n=30 then the T-total is 5x30+7=157 and 30+31+32+23+41=157.
N=65 so 5x65+7=332 and 65+66+67+58+76=332. Now I will check the formula on an 8 by 8 grid.
N=9 so the formula is 5x9+7=52 and the T-total is 9+10+11+3+19=52
N=13 so the formula is 5x13+7=72 and 13+14+15+7+23=72.
I have now proved that my formula works on a 9 by 9 grid as well so now I shall try on a 10 by 10 grid.
N=11 the formula is 5x11+7=62 and 11+12+13+3+23=62
N=32 the formula becomes 5x32+7=167 and the T-total is 32+33+34+24+44=167
N=17 5x17+7=92 and 17+18+19+29+9=92
I have now proved that my formula works on a 9 by 9 grid and a 10 by 10 now I will try it on an 8 by 8 and see if there is a link to the other formula.
N=9 so the formula becomes 5x9+7=52 and 9+10+11+19+3=52
N=29 so the formula becomes 5x29+7=152 and 29+30+31+23+39=152
N=49 so the formula this time is 5x49+7=252 and the t-total for this shape is 49+50+51+43+59=252 I have now proved that my formula works on a 9 by 9 grid an 8 by 8 grid and a 10 by 10 grid so now I am going to see if there is a link between the two formulae I have.
Normal= 5n-7g Rotated 90o= 5n+7
To see a link I need one more formula so I will work out the formula for a shape rotated by 180o.
This become equal to 5n+7g I am going to test this on 2 shapes from each grid size to check if it works.
First I will try 8 by 8.
34x5+7x8=226 and the sum of T-shape is 34+42+40+49+51=22
22x5+7x8=166 and the T-total is 22+30+38+37+39=166
Now I will try it 9 by 9.
2x5+9x7=73 and 2+11+20+21+19=73
5x34+9x7=233 and 34+43+51+52+53=233 so the formula works on 9 by 9 I will now try it on 10 by 10.
5x54+7x10=340 and 54+64+74+73+75=340
5x78+7x10=460 78+88+98+99+97=460 I have now proved that my formula works on all sized gids so now I shall compare the three formulae I have.
Normal=5n-7g rotated 90o clockwise=5n+7 rotated 180o=5n+7g
I predict that 270 is 5n+7. I will now check.
5n-7so I was correct.
