= 10
As illustrated the product difference didn’t change.
At this stage I made an hypothesis and stated that wherever the 2x2 box was drawn within the 10 x 10 grid the product difference wouldn’t vary.
To verify this claim, I tested another three boxes. I would use the same technique throughout ie. the same as above.
1.
Product difference = (10 x 19) – (9 x 20)
= 10
2.
Product difference = (38 x 47) – (37 x 48)
= 10
3.
Product difference = (64 x 73) – ( 63 x 74)
= 10
My above results verify my hypothesis. The product difference, when using a 2x2 box drawn within a 10x10 number grid is always 10.
So far my results were found using only 2x2 boxes. So to investigate further, I varied the size of the box to 3x3 and calculated the product difference.
Product difference = (16 x 34) – (14 x 36)
= 40
The product difference was 40. So again I hypothesized that wherever the box was drawn within the 10x10 grid the product difference would be 40.
To verify this claim, I chose another three boxes. I used the same techniques as previously.
1.
Product difference = (45 – 63) - (43 x 65)
= 40
2.
Product difference = (29 x 47) - (27 x 49)
= 40
3.
Product difference = (73 x 91) – (71 x 93)
= 40
Again my hypothesis was verified (Absolute proof would involve checking every possible combination).
I carried on using 4x4, 5x5, 6x6 and 7x7 boxes. And decided that the product difference were always constant.
To show my results in a clearer way here is a table.
In an effort to find a pattern I wrote the product differences in sequence form.
10 40 90 160 250 360
↓ ↓ ↓ ↓ ↓
First difference 30 50 70 90 110
↓ ↓ ↓ ↓
Second difference 20 20 20 20
This illustrates that the second difference is constant. From this, I could establish a formula. I would use the Nth term technique.
When forming a formula in this way, firstly I have to place the letter N after the constant, but in this case the constant is the second difference in the sequence, therefore I must use 20 N . Another rule when using the second constant is to half it. Therefore the formula would be 10N .
To test this I drew a table.
Therefore the formula is simply 10N .
At this stage I made another hypothesis and stated that the 10 in the formula linked directly to the grid size ie. 10x10.
So if I drew a 7x7 grid the formula to find the product difference would be 7N .
To verify this I drew different size boxes within the 7x7.
2x2 Boxes
1.
(2x8) – (1x9)
Product difference = 7
2.
(5x11) – (4x12)
Product difference = 7
3x3 Boxes
1.
(7x19) – (5x21)
Product difference= 28
2.
(35x47) – (33x49)
Product difference = 28
4x4 Boxes
1.
(25x43) – (22x46)
Product difference = 63
2.
(28x46) – (25x 49)
Product difference= 63
5x5 Boxes
1.
(5x29) – (1x33)
Product difference = 112
2.
(19x43) – (15x47)
Product difference= 112
Once again to establish a formula I wrote the product differences I sequence.
7 28 63 112
↓ ↓ ↓
First difference 21 35 49
↓ ↓
Second difference 14 14
To obtain my formula I must use N as I’m using the second difference, also as explained previously half the constant must be used, therefore the formula would be 7N .
Thus verifying my previous hypothesis.
Conclusion
I can now safely say that to predict a product difference I could use this formula:
Product difference=
Grid size (ie. 10x10 would be 10, 7x7 would be 7) x Nth term(Nth term being 2x2 box would be 1, 3x3 box would be 2)
Eg.
To predict the Product difference of a 3x3 box within a 10x10 grid:
=10N
= 10 x 3
= 90
Evaluation
Although my formula is quite efficient, I have found that it doesn’t work for irregular size boxes ie. 2x3 or 3x4 etc. If I had extra time I could consider investigating this problem further eg adapting the formula, maybe factorizing it.