Question 65
Consider the infinite quadrant in the figure, where all the small squares have side $1$. Is it possible to color some of the squares black so that both of the following properties are satisfied?
- For each natural number $n$, the square with vertex at $O$ and side $n$ (with sides parallel to the axes) has a greater number of black squares than the number of white squares.
- On every infinite diagonal of small squares parallel to the one shown in the figure, there are at most a finite number of black squares.
- For each natural number $n$, the square with vertex at $O$ and side $n$ (with sides parallel to the axes) has a greater number of black squares than the number of white squares.
- On every infinite diagonal of small squares parallel to the one shown in the figure, there are at most a finite number of black squares.