To solve this problem I have broken down the table as much as possible starting with the first query given in the problem.Ex (1) first boxHighlighted box: the product of the top right hand number and the bottom left handnumber minus the product of the top left hand number and the bottom right hand number2*13= 2612*3= 3636-26= 10Ex (2) second box45*56=252055*46= 25302530-2520= 10The 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 followsThe 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) = 10I tried using the formula for what I have found before in table Fig. B below but this time I added more columns to the problem123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778708081828384858687888990919293949596979890100Fig.
BThe first column grid only 7 columns were highlighted and the following was taken into consideration6 first number is n7 the second number is n +1By trying to find the formula for the number of columns used I introduced c in the formulaTherefore when 7 columns were usedThe number below n =n+cThe number next to is as n +c=1Using the formula I found6 (n), 7(n+1)13 (n+c), 14(n+c+1)6*14=8413*7=9191-84=7When 6 columns were used55(n), 56(n+1)61(n+c), 62(n+c+1)55*62= 341056*61= 34163416-3410=6123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778708081828384858687888990919293949596979890100Fig. CIn Fig. C I observed the followingIn 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=4511+12+13+14+15+16+17+18+19=135135-45= 90Second boxThis time only eight columns where highlighted81+82+83+84+85+86+87+88= 67691+92+93+94+95+96+97+98= 756756-676= 80The 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.