Let $\mathbb{P}=\{0,\dots,8\}$ be the set of $9$ points in a coordinate geometry over a field $(K,+,\cdot)$. Let moreover consider the set of lines in this geometry:
$$\mathbb{B}=\{\{0,1,2\}, \{3,4,5\}, \{6,7,8\}, \{0,3,6\}, \{1,4,7\}, \{2,5,8\}, \{0,4,8\}, \{3,7,2\}, \{1,5,6\}, \{2,4,6\}, \{0,5,7\}, \{1,3,8\}\}$$
The set above represents all the lines over the square below.
If one plays the lottery giving all the sheets of $\mathbb{B}$ he will be surely guess $2$ numbers of the $3$ extracted.
Claim: Any other set $\mathbb{B}$ with the above characteristic has still $12$ elements.
How to prove the claim? I know that this geometry is isomorphic to the one on $GF(3)$, does it helps?