A \(3\times 3\) square is filled with the numbers \(-1, 0, +1\). Prove that two of the 8 sums in all directions – each row, column, and diagonal – will be equal.
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).
Is it possible to fill an \(n\times n\) table with the numbers \(-1\), \(0\), \(1\), such that the sums of all the rows, columns, and diagonals are unique?
Fill the free cells of the “hexagon” (see the figure) with integers from 1 to 19 so that in all vertical and diagonal rows the sum of the numbers, in the same row, is the same.
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
Find out the principles by which the numbers are depicted in the tables (shown in the figures below) and insert the missing number into the first table, and remove the extra number from the second table.