d = 20
From this investigation I have found that if you multiply the diagonally opposite corners in a 2 x 3 grid from anywhere in the square grid, the difference will be 20.
2 x 4 Grid
d = 30
d = 30
d = 30
From my results, I have discovered that if you multiply diagonally opposite corners of a 2 x 4 box grid, taken from any position in a 100 square grid, the difference between the two answers will be 30.
2 high boxes
As you can see in my table, I feel I could now use my findings to predict the difference for a 2 x 5 grid. As I have already predicted, the difference will be 40.
2 x 5 grid
d = 40
My prediction was correct in saying that the difference in a 2 x 5 grid would be 40.
I have noticed that it does not make any difference to the outcome wherever the grid is taken from in a 100 square grid.
From now on I only need to try one box of each size as I know that position has no effect on the difference.
2 x w Algebra (Any width)
(n) (n+w+9) = n2 + nw +9n
(n+10) (n+w-1) = n2 + nw – 1n +10n + 10w -10
(n2 +nw+9n) – (n2+nw+9n+10w-10) = d
10w-10= d
I will now do an example to show how my formula works:
If the width is 3:
10w-10= d
(10x3)-10= d
30-10=20
d=20
As is shown below, this is the same answer I got for a 2x3 grid:
d = 20
I will now investigate to find out if there is a pattern for the height changing in a grid. I will try a series of 3 high boxes.
3 High Boxes
3 x 2
3 x 3
3 x 4
From my investigation into 3 high boxes, I feel I could now predict the difference in a 3 x 5 box. I predict that the difference in a 3 x 5 box will be 80.
I say this because over the past few boxes, the difference has increased by 20 when the width has increased by 1.
3 x 5
My prediction was correct in saying that the difference in a 3 x 5 grid would be 80.
I will now increase the height to 4 to see if a pattern emerges.
4 High Boxes
4 x 2
4 x 3
4 x 4
I have noticed that the difference increases by 30 when you increase the width of the box by 1. I believe I could now predict the difference in a 4 x 5 box. I predict the difference in a 4 x 5 box will be 120.
4 x 5
My prediction was correct in saying that the difference would be 120.
Tables
2- High 4-High
3-High
5-High
Since I went up to 5 x 6 in the table, I will try a 5 x 6 box to test my prediction (200).
5 x 6
My prediction was correct.
Algebra for h x w grid in any position on a g x g grid
gh-g+n gh-g+n+gw-1
n(gh-g+n+w-1)
=ngh-ng+(n2)+nw-n
(n+w-1)(gh-g+n)
=ngh-ng+(n2)+wgh-wg+wn-gh+g-n
d=(ngh-ng+(n2)+nw-n)-(ngh-ng+(n2)+wgh-wg+wn-gh+g-n)
d=wgh-wg-gh+g
I will now demonstrate how my formula works:
If g=10 h=4 w=3 n=5:
d= wgh-wg-gh+g
d=(3X10X4)-(3x10)-(10x4)+10
d=120-30-40+10
d=60
As you can see below, this is the answer I got for a 4x3 grid.
By Philip Redpath
Algebra for h x w box on a 10x10 grid
10h-10+n 10h-10+n+w-1
n(10h-10+n+w-1)
=10hn-10n+(n2)+wn-n
(n+w-1)(10h-10+n)
=10hn-10n+(n2)+10wh-10w+nw-10h+10-n
d=(10hn-10n+(n2)+wn-n)-( 10hn-10n+(n2)+10wh-10w+nw-10h+10-n)
d=10wh-10w-10h+10
I will now demonstrate how my formula works:
h=3 w=2 n=6
d=10wh-10w-10h+10
d=(10x2x3)-(10x2)-(10x3)+10
d=60-20-30+10
d=20
As you can see below, this is the same answer I got for a 3x2 grid:
