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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 $R_1=y$, $R_2=x$.

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Task 1 was answered. Task 2:

Not possible. We always have $\begin{bmatrix}R_1 \\ R_2\end{bmatrix} = M\begin{bmatrix}x \\ y\end{bmatrix}$ for some matrix $M$ that depends only on the sequence of operations. Initially $M = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, and the given operations cause $M$ to be left-multiplied by $\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$, $\begin{bmatrix}1 & 0 \\ -1 & 1\end{bmatrix}$, $\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}$, and $\begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}$ respectively. All of these matrices have determinant $1$, so we will always have $\det M = 1$, which makes it impossible to reach the desired $\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$ with determinant $-1$.

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  • $\begingroup$ I had a feeling this was true. Very neat proof. $\endgroup$ – Dr Xorile Jan 3 at 15:23
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Task 1

1. $R_1 - R_2\to R_1$ (now $R_1 = x-y$, $R_2 = y$)
2. $R_1 + R_2\to R_2$ (now $R_1 = x-y$, $R_2 = x$)
3. $R_1 - R_2\to R_1$ (now $R_1 = -y$, $R_2 = x$)

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    $\begingroup$ Task 2 the second operation isn't in the list of allowed operations $\endgroup$ – Dr Xorile Jan 3 at 6:26
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    $\begingroup$ @DrXorile Oops, so it seems... $\endgroup$ – jafe Jan 3 at 6:27

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