I came up with this problem inspired by the limitations of an old non-scientific calculator I owned years ago (the two registers were the display, and an internal memory for an additional number).
We have a primitive calculator with only two registers $R_1$ and $R_2$, and the following four operations:
$R_1+R_2\to R_2$ (add the content of register $R_1$ to register $R_2$.)
$-R_1+R_2\to R_2$ (subtract the content of register $R_1$ from register $R_2$.)
$R_1+R_2\to R_1$ (add the content of register $R_2$ to register $R_1$.)
$R_1-R_2\to R_1$ (subtract the content of register $R_2$ from register $R_1$.)
For instance, if $R_1=x$ (register $R_1$ contains number $x$) and $R_2=y$ ($R_2$ contains $y$), after applying the operation $R_1+R_2\to R_2$ we end up with $R_1=x$ and $R_2=x+y$.
Assume that initially we have $R_1=x$ and $R_2=y$, where $x$ and $y$ are arbitrary numbers. For each of the following tasks describe a sequence of operations that would allow us to perform it, or prove that the task cannot be performed (task 1 is very easy, the real challenge is about task 2):
Task 1. Swap the contents of registers $R_1$ and $R_2$ changing the sign of $y$ in the process, so we would end up with $R_1=-y$, $R_2=x$.
Task 2. Swap the contents of registers $R_1$ and $R_2$, so that we would end up with $R_1=y$, $R_2=x$.