= Solution
This numbered item specifies the model used below. A convenient <sample space> is the collection of $36$ ordered pairs $(s_1,s_2)\in\{1,\ldots,6\}^2$, each with <probability> $1/36$. The corresponding <random variables> are determined by the sum and difference. Inverting the two equations gives
$$
\boxed{S_1=\frac{X+Y}{2},\qquad S_2=\frac{X-Y}{2}.}
$$
Thus a candidate pair $(X,Y)=(i,j)$ is possible exactly when $(i+j)/2$ and $(i-j)/2$ are both <integers> between $1$ and $6$. In particular, $i$ and $j$ must have the same parity. This inverse description will determine the <joint probability mass function>.
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