8. Fill the given grid with numbers 1 to 9. Make sure that each
row and the two diagonals have the same total.
4
5
6
Answers
Denote by A, B, C - the numbers in the first row, by D, E, F - the numbers in the second row, and by G, H, I - the numbers in the third row (from left to right in each row).
A is the least number in the table, therefore, A = 1.
I is the greatest number in the table, therefore, I = 9.
There are 4 numbers (A, B, C, D) which are less than E, and there are 4 numbers (F, G, H, I) which are greater than E. Therefore, E = 5.
Furthermore, the set {B, C, D} = {2, 3, 4} and B < C. To determine the values of these variables it is sufficient to determine the value of D (total of 3 variants). Then B will be the minimal element of the set {2,3,4}\{D} and C will be the maximal element of this set.
Similarly, the set {F, G, H} = {6, 7, 8} and G < H. To determine the values of these variables it is sufficient to determine the value of F (total of 3 variants). Then G will be the minimal element of the set {6, 7, 8}\{F} and H will be the maximal element of this set.
Finally, the total number of ways to fill the table is equal to 3 x 3 = 9.