I put my results into table and observed that the difference is square number because the box and the grid are square shaped.
I predict that for a 5 by 5 square the difference between the products will be 160.
46 x 90 = 4140 50 x 86 = 4300 4300 - 4140 = 160
I will now attempt to prove my results algebraically.
After doing this I will check my answer using the nxn formula
2x2 square
n n+1
n+10 n+11 (24x33)- (23x34)
792-782=10
(n+1) (n+10)-n (n+11)
n2 +10n + n+10 - n2 +11n n2 +11n +10 - n2 +11n =10
In both ways the difference is
5x5 square
n n+4
n+40 n+44
n (n+44) - [(n+4) (n+40)] (5x41)-(1x45)
n2+44n - n2+40n + 4n +160 205-45= 160
n2+44n +160 - n2+44n =160
In both ways the difference is 160
After doing this I will check if the nxn formula will work in a bigger grid size like 11x11
n n+1
n+11 n+12
(n+11)x(n+1)-n (n+12) (2x12)-(1x13)
n2+n+11n+11- n2+12 24-13=11
n2+12n+11- n2+12
n2+12n+11- n2+12=11
In both ways the difference is 11
Now I will try to find another way to see if there is any rule or pattern, which connects the box size in another way to find the difference.
If you minus one from the boxed squared size (B) and then time it by the grid size (G) it will give me the difference (D).
(B-1) 2 x G= D D = Difference G = Grid size B = box squared size
For example if the box size is (4x4)
(B-1) 2 x G = D B=4 G = 10 D = 90
(4-1) 2 x 10 =90
For example if the n th term is 6 (7 x 7) the difference would be 360.
(B-1) 2 x G= D B = 7 G = 10 D = 360
(7-1) 2 x 10 = 360
I will now try a larger grid size, which is 11 by 11.
3 x 15 = 45 4 x 14 = 56 56 - 45 = 11
G = Grid D = difference B = box squared size
(B-1) 2 x G = D B = 2 G = 11 D = 11
(2-1) 2x 11= 11
40 x 64 =2560 42 x 62 =2604 2604 - 2560 = 44
(B-1) 2 x G = D B= 3 G = 11 D = 44
(3-1) 2 x 11 = 44
Having checked a further series of results I found that even for a larger grid the same formula can work for square only it does not matter the size of the square.
If I were to consider rectangles the above formula will not work for rectangles so I will try another formula and see if it is going to work with squares too.
I am going to do a table and the grid size will be 10 by 10.
1 2 x 24 = 288 14 x 22 = 308 308 - 288 = 20
S= SIZED G = GRID D = DIFFERENCE
S-1 x G = D
Sized = 2 x 3 G = 10 D = 20
2-1 =1
3-1 =2
1 x 2 =2
2 x 10 =20
37 x 60 =2220 40 x 57 = 2280 2280 - 2220 = 60
S= SIZED G = GRID D = DIFFERENCE
S-1 x G = D
SIZED = 3 x 4 G = 10 D = 60
3-1 = 2
4-1 = 3
2 x 3 = 6
6 x 10 = 60
I could prove that this formula can work for both rectangle and square.
53 x 64 = 3392 54 x 63 = 3402 3402 - 3392 = 10
S= SIZED G = GRID D = DIFFERENCE
S-1 x G = D
S = 2 x 2 G = 10 D = 10
2-1 = 1
2-1 = 1
1 x 1 = 1
1 x 10 = 10
FINALLY
In this project I found that not all formula can work for rectangle and square.
But I found a formula can work for rectangle and square it does not matter the size of the box and I will proof that it does not matter for grid sizes too.
28 x 54 = 1512 30 x 52 = 1560 1560 - 1512 = 48
S= SIZED G = GRID D = DIFFERENCE
S-1 x G = D
S = 3 x 3 G = 12 D = 48
3-1 = 2
3-1 = 2
2 x 2 = 4
4 x 12 = 48
If I were to extend this project further I would try and do cube in three dimensions.
