I was watching a game show which fills in a 5x5 "bingo"-style grid, one element at a time. At the end of the game show, contestants try to get a bingo (that is, 5 filled-in elements in a row either horizontally, vertically, or diagonally). The grid starts with 12 squares filled in, and, based on the contestants' previous performance, they draw randomly to fill in the remaining squares.

Contestants who do reasonably well get around 8 draws. I was wondering if, at some point, these extra draws become unnecessary. The rules of the game cap the number of draws at 10, leaving at least 3 grid elements unfilled. To put it another way, what is the minimum number of elements on a 5x5 grid that need to be filled in to ensure that there will be a bingo?

I know that 20 isn't quite enough, from the following counterexample:

Code:
- x x x x
x - x x x
x x - x x
x x x - x
x x x x -

x = filled-in
- = open
[I posted this without trying to analyze it myself since I was thinking somebody might be interested in it as a puzzle.]