The product of the top right hand number and the bottom left hand number in each square of numbers minus the product of the top left hand number and the bottom right hand number regardless of the position of the square is always equal to ten (10).
From this I tried to get a formula for the nth term as follows
The first number (2) is n and the second number 3 is n+1,
The number below n (12) is n+10 and the one next to it (13) is n+11.
n(n+11) =
n+10(n+1)=
n+10(n+1) – n(n+11) = 10
I tried using the formula for what I have found before in table Fig. B below but this time I added more columns to the problem
Fig. B
The first column grid only 7 columns were highlighted and the following was taken into consideration
6 first number is n
7 the second number is n +1
By trying to find the formula for the number of columns used I introduced c in the formula
Therefore when 7 columns were used
The number below n =n+c
The number next to is as n +c=1
Using the formula I found
6 (n), 7(n+1)
13 (n+c), 14(n+c+1)
6*14=84
13*7=91
91-84=7
When 6 columns were used
55(n), 56(n+1)
61(n+c), 62(n+c+1)
55*62= 3410
56*61= 3416
3416-3410=6
Fig. C
In Fig. C I observed the following
In the first box (top) I highlighted nine (9) columns and decided to add all the top numbers and all the bottom numbers to get the difference.
1+2+3+4+5+6+7+8+9=45
11+12+13+14+15+16+17+18+19=135
135-45= 90
Second box
This time only eight columns where highlighted
81+82+83+84+85+86+87+88= 676
91+92+93+94+95+96+97+98= 756
756-676= 80
The difference between the sum of the top numbers and bottom numbers in each column grid is the amount for columns in the grid by 10.
Considering what I found initially:
Investigation 1:
The product of any square regardless of it’s location in the grid is 10.
Investigation 2:
n+c+1 (c being the number of columns) the answer is always equal to the number of columns used in the grid.
Investigation 3:
The total of numbers in the top row of a grid subtract the total of numbers below is always equal to the number of columns used in the grid multiply by ten (10) which is the product of any square.
In conclusion I found that the general formula for a grid with c columns multiply by the general formula in a square is equal to the sum of numbers in the grid square and grid column.
