Mathematics of SET game

Source: Sent to me by Pritish Kamath (


Have you ever played "SET"? You have to play it.

Even if you have not played the game, the game can be stated in a more abstract way as follows:
There are 12 points presented in F34 and the first person to observe a "line" amongst the 12 given points gets a score. Then the 3 points forming the line are removed, and 3 random fresh points are added.

Problem 1: How many points in F34 are needed to be sure that there exists a line among them?
Problem 2: Given 12 random points in F34, what is the probability that there exists a line among them?

We have not solved the problem yet. It can be very difficult or very easy.

Update (22/12/14):
It turns out to be a very very difficult problem.
The answer to Problem 1 is 21.