This can be considered both a puzzle and a solitaire game.
Make an n-by-n grid on paper. (I suggest that n = 3 for beginners.)
Fill the grid with UNIQUE positive integers, one integer per square of the grid. The numbers need not be consecutively valued necessarily.
Your score is the largest integer m such that integers 1 through m all occur as sums within the grid (without missing any positive integers <=m).
A "sum" is of any number of addends (possibly just 1) that are all *consecutively placed* within a row of the grid or a column of the grid. (No diagonals in this variation.)
So, for example, if we have the following 3-by-3 grid:
2 9 8
3 1 10
7 6 15
...the sums 1 through 19 all occur in this grid. So, I get a score of 19.
(Notice that some values of sums occur more than once.)
I bet it is easy to do better than I did with a 3-by-3 grid, since my grid is inefficient, and I didn't try too hard to find it.
(Note: There are 27 possible sums in a 3-by-3 grid of 1, 2, or 3 consecutively placed addends. I don't know what the largest possible score is for a 3-by-3 grid, though.)
You can "play" someone else by both of you trying to score as well as you can on same-sized grids. Just try to outscore your opponent.
Update: The largest score I personally received on the 3-by-3 grid is 20. But someone using a computer, I think, found a 3-by-3 grid with a score of 25.
Update2: Someone else found a 3-by-3 grid with a score of 26.