You have an n × n grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive i+j points if there were i crosses in the same row and j crosses in the same column before you placed the new cross. Which are the possible total scores you can get?
I think there is only one total possible no matter how you fill:
I look at it this way: The order in which the cells of a row are filled is immaterial. Always you will gain 0, 1, 2, ..., n-1 game points with each successive fill-in (not necessarily made on successive turns), counting the growing number of elements in that row. So, by the end, each row contributes sum(n-1) = (n-1) n/2. Row and column contributions build independently but increase in the same way. Since there are 2n rows and columns: Total = (2n) (n-1) n/2 = (n-1) n^2
Edited on December 7, 2020, 9:46 am