Number Grid.

Example 1

(1 x 23) - (3 x 21) = 40

Example 2

(53 x 75) - (55 x 73) = 40

Example 3

(78 x 100) - (80 x 98) = 40

Thus proving that 40 is the continuous answer when the sum is completed within a 3 x 3 grid.

I then increased the size of the cell selection to a 4 x 4 area, and my results were as follows:

Example 1

(2 x 35) - (5 x 32) = 90

Example 2

(61 x 94) - (64 x 91) = 90

Example 3

(44 x 77) - (47 x 74) = 90

Thus proving that 90 is the constant answer when the sum is completed within a 4 x 4 grid.

I continued to expand the grid size until I was confident that the results would always follow the same pattern, when the sum was attempted within the 10 x 10 matrix. My overall results were as follows:

Grid Size

Product Difference

2 x 2

0

3 x 3

40

4 x 4

90

5 x 5

60

6 x 6

250

As my answers thus far appeared to be following a pattern, I decided to see if I could find a method for finding the product difference by using an NTh. term formula.

The formula that I found effective was this:

Total grid size x NTh term 2 = product difference

i.e.

NTh. term

st

2nd

3rd

4th

5th

Grid size

2 x 2

3 x 3

4 x 4

5 x 5

6 x 6

NTh. term calculation

0n2

Product difference

0

40

90

60

250

Note that the NTh. term is always 1 less than the grid size

To see if this formula would work with a different size number grid I used the same formula as before only this time worked from a 10 x 8 matrix. Within the grid I tried the same variations as above i.e. 2 x 2, 3 x 3 etc. and discovered that the results remained exactly the same.

I continued to investigation further, only this time I experimented using the 8 x 8 grid that can be seen below:

When calculating the product from the above selection I encountered different results than found using the 10 x 10 matrix. For example, when working with a 2 x 2 grid:

Example 1

(12 x 21) - (13 x 20) = 8

Example 2

(28 x 37) - (29 x 36) = 8

Example 3

(55 x 64) - (56 x 63) = 8

I found that the product differences remained the same however, in an effort to once again, further the investigation I carried out several more individual tests on each of the grid selection sizes used earlier and received the following results.

Grid Size

Product Difference

2 x 2

8

3 x 3

32

4 x 4

72

5 x 5

28

6 x 6

200

I then decided to try out the NTh term formula I used earlier to see if it was effective when using an 8 x 8 matrix and found that the formula once again delivered the correct results. i.e.

Total grid size x NTh term 2 = product difference

NTh. term

st

2nd

3rd

4th

5th

Grid size

2 x 2

3 x 3

4 x 4

5 x 5

6 x 6

NTh. term calculation

8n2

Product difference

8

32

72

28

200

The formula was also proved successful when calculating the product differences on other number grid matrices of differing sizes.

I then decided to find out if a rule applied when working with a 3 x 2 number grid within a 10 x 10 matrix.

Again, I multiplied the top left number with the bottom right and then multiplied the top right number with the bottom left and subtracted my results to give the product differences.

I experimented with three individual 3 x 2 grids to prove my results remained constant.

Example 1

(12 x 24) - (14 x 22) = 20

Example 2

(44 x 56) - (46 x 54) = 20

Example 3

(88 x 100) - (90 x 98) = 20

I found that the product differences remained the same however, in an effort to once again, further the investigation I carried out several more individual tests, increasing the size of the internal grid on each occasion. My results were as follows:

Grid Size

Product Difference

3 x 2

20

4 x 3

60

5 x 4

20

6 x 5

200

7 x 6

300

And again, I worked out a formula to find the product difference.

Formula: Grid size x internal grid length -1 x internal grid height -1

or: 10(L-1)(H-1)

NTh. term

st

2nd

3rd

4th

5th

Grid size

3 x 2

4 x 3

5 x 4

6 x 5

7 x 6

NTh. term calculation

0(L-1)(H-1)

Product difference

8

32

72

28

200

Note that the NTh. term is always 1 less than the grid height