Bingo variant- Minimum no of boxes

Question

We play $5$ line bingo but in a different way. We take a $5×5$ grid, fill it randomly with numbers from $1$ to $25$. Then the players call out numbers they want and when you get $5$ full lines crossed out (including vertical,horizontal or/and diagonal) you win the game.

I was wondering what would be the minimum number of boxes a player needs to cross out to win the game.

By figuring it out manually, I believe it would be $16$. Similarly for a $4×4$ grid it would be $11$ and for a $6×6$ grid it would be $24$. (For an $n × n$ grid you need $n$ full lines crossed out to win)

Is there some way to do this mathematically? And can we extend this to other $n × n$ grids?

If there is a way, will it change depending on whether n is even or odd because of the nature of the diagonal line intersecting and not intersecting in case of odd and even grids respectively? Perhaps for even grids the series resembles A293414 https://oeis.org/A293414

Answer 1

For the $5\times 5$ grid, the minimum is $16$:

o o o o o
. o . o .
. o o . .
. o . o .
o o o o o

For grid sizes $1$ thru $12$: $$1,3,6,11,16,24,32,43,54,68,82,99$$ Possibly related to OEIS A087099