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Tic tac toe points (Posted on 2020-12-07) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Danish Ahmed Khan    
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soln | Comment 1 of 3
I think there is only one total possible no matter how you fill:
(n-1) n^2 

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
  Posted by Steven Lord on 2020-12-07 09:07:19

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