Kanel-Belov A.Y.

On a plane, there is a square and a point $P$ is marked with invisible ink. A person wearing special glasses can see the point. If a line is drawn, they answer the question of which side of the line $P$ lies on (if $P$ lies on the line, they say that $P$ lies on the line).

What is the minimum number of such questions that need to be asked to determine whether the point $P$ lies inside the square?
Let our square be $A B C D$. First, we draw the diagonal AC. If it turns out that $P$ lies on the same side of it as vertex $B$, it will be enough to find out on which side of the lines $A B$ and $B C$ the point $P$ lies. The case where the person indicates the half-plane containing point $D$ in response to the first question is analogous. If only one or two questions are asked, fewer than three lines are drawn. No matter what the answers are (with the exception of the case where the point lies at the intersection of two drawn lines), we can only find out whether the point belongs to the parts into which the plane is divided by the drawn lines. But these parts are not bounded, and belonging to them cannot be proof that the point belongs to the square.

## Answer

Three questions.