I found an interesting problem: We are given $n$ bits in a row, some of them are equal to $1$, and some of them are equal to $0$. This is our starting combination. And we are also given the ending combination. The only operation we can perform is negation of chosen bit. But it negates also its neighbours (first and last bits have only one neighbour). Our task is to find the minimal number of operations we need to perform in order to obtain ending combination from starting combination. Or say it's not possible. By this I mean to write an effective algorithm solving this problem.
1) start: $1001$; end: $0010$. Optimal number of operations is $2$, because $1001\rightarrow 0101\rightarrow 0010$ (firstly negate first bit, then third).
2) But from: $10$; to: $00$ it is impossible to get.
I think it's a quite well known problem but I'm not sure. Maybe it's known in some other form. I would be very grateful for any hints.