On an $n \times n$ chessboard, we write the number 1 in the first field of the first row, and in the remaining fields of the first row and the first column, we write the numbers $2,3, \ldots, n$ respectively. In every other field, we write the number that is one less than the sum of the numbers at the beginning of the row and the column. Then, we place a point mass of the written number grams at the center of each field. Where is the center of mass of this point system? (We consider the chessboard to be weightless.)

Where is the center of mass if we fill the inner fields of the table with the product of the two numbers instead of the reduced sum?