2. Given is a square $A B C D$ with side length 4. Determine the largest natural number $k$ such that, for any arrangement of $k$ points strictly inside the square $A B C D$, there always exists a square with side length 1, contained within the square $A B C D$ (whose sides do not have to be parallel to the sides of the square $A B C D$), in whose strict interior none of the observed $k$ points lie. (Bojan Bašić)
2. The answer is $k=15$.

It is clear that $k=15$ satisfies the conditions. Indeed, divide the square $A B C D$ into 16 unit squares; no matter how we arrange 15 points inside the square $A B C D$, at least one of these unit squares will not contain any of these points.

Let's show that $ka$, then it contains at least one marked point.