Consider a finite number of lines in the plane, none of which are parallel, and no three of which are concurrent. These lines divide the plane into several regions—both finite and infinite.

### Problem Analysis

In this scenario, we assign the number $1$ or $-1$ to each of the finite regions. The operation allowed involves selecting any triangle formed by the intersection of any three of these lines and flipping the sign of the numbers in the regions within this triangle. The problem asks us to determine if it is possible to make all finite regions contain the number $1$ using a sequence of these operations, starting from an arbitrary distribution of $1$s and $-1$s.

### Insight into the Problem

The key observation is related to the parity (evenness or oddness) of the number of $-1$s in the finite regions:

1. Invertibility of Operations: Since each operation affects exactly three regions (the interior of the chosen triangle), it converts the signs of these regions from $1$ to $-1$ or vice versa. Thus, each operation switches the parity of the number of $-1$s among these three regions: if the number was even, it becomes odd, and vice versa.

2. Parity Consideration: Consider the sum of numbers over all finite regions modulo 2. This sum effectively tracks the parity of the $-1$s. Hence, if the initial configuration has an odd number of regions with $-1$, there will always remain an odd number of regions with $-1$. Conversely, if the initial configuration has an even number of $-1$s, this parity remains unchanged after any operation.

### Conclusion

Let us analyze whether it is always possible to turn all finite regions into $1$:
- If initially the parity of the number of regions with $-1$ is even, it might be possible to convert all $-1$s to $1$ since the parity allows achieving zero $-1$s.
- However, if the initial configuration starts with an odd count of $-1$s, flipping groups of three $-1$s (or their absence) will always leave an odd number of $-1$s in the plane.

Given no control exists over the initial parity configuration, there is no guarantee of achieving a plane filled entirely with $1$, particularly when starting with an odd number of regions labeled $-1$.

Thus, the answer to the problem is:
$$\boxed{\text{No}}$$
This indicates it is not always possible to reset all regions to $1$ using the given operations, corresponding to potentially insolvable initial configurations with odd $-1$ counts.