Bakayev E.V.

Petya drew a square on the plane, divided it into 64 identical smaller squares, and colored them in a checkerboard pattern with black and white colors. After that, he thought of a point strictly inside one of these smaller squares. Vasya can draw any closed broken line without self-intersections on the plane and get an answer to the question of whether the guessed point is strictly inside the broken line or not. What is the minimum number of such questions Vasya needs to determine the color of the guessed point - white or black?
To determine the color of a point in one question, a broken line is needed, relative to which all squares of one color lie inside, and the other - outside. But then it contains all segments separating adjacent ones.

Let's show how to guess the color of a point in two questions. We will construct a broken line, inside which all the odd horizontal rows of the square will fall and only they (see fig.).

![](https://cdn.mathpix.com/cropped/2024_05_06_123645a38d760684829ag-31.jpg?height=512&width=594&top_left_y=104&top_left_x=732)

Then with the first question, we will find out the parity of the horizontal row containing the guessed point. With a similar second question, we will find out the parity of the vertical row containing it. It remains to note that the color of the square is determined by the parity of the sum of its "coordinates".

## Answer

In two questions.