Question 31
We consider in the plane a finite number of lines such that there are no two parallel nor three concurrent. These lines divide the plane into finite and infinite regions. In each finite region a number $1$ or a number $-1$ is written. The permitted operation consists of choosing any triangle formed by three of the considered lines (this triangle can be crossed by other of the considered lines), and multiplying all the numbers contained in this triangle by $-1$.
Determine whether it is always possible, through operations of this type, to obtain a $1$ number in all regions, no matter what the initial distribution of the $1$ and $-1$ numbers is.
Determine whether it is always possible, through operations of this type, to obtain a $1$ number in all regions, no matter what the initial distribution of the $1$ and $-1$ numbers is.