The d-distance p-packing domination number \(\gamma _d^p(G)\) of a graph G is the cardinality of a smallest set of vertices of G which is both a d-distance dominating set and a p-packing. If no such set exists, then we set \(\gamma _d^p(G) = \infty \) . For an arbitrary strong product \({G\boxtimes H}\) it is proved that \({\gamma _{d}^p(G\boxtimes H) \le \gamma _d^p(G) \gamma _d^p(H)}\) . By proving that \(\gamma _{d}^{p}(P_m \boxtimes P_n) = \Big \lceil \frac{m}{2d+1} \Big \rceil \Big \lceil \frac{n}{2d+1} \Big \rceil \) , and that if \(\gamma _{d}^{p}(C_n) < \infty \) , then \(\gamma _{d}^{p}(P_m \boxtimes C_n) = \Big \lceil \frac{m}{2d+1} \Big \rceil \Big \lceil \frac{n}{2d+1} \Big \rceil \) , the sharpness of the upper bound is demonstrated. On the other hand, infinite families of strong toruses are presented for which the strict inequality holds. For instance, we present strong toruses with difference 2 and demonstrate that the difference can be arbitrarily large if only one factor is a cycle. It is also conjectured that if \(\gamma _d^p(G) = \infty \) , then \(\gamma _d^p(G\boxtimes H) = \infty \) for every graph H. Several results are proved which support the conjecture, in particular, if \(\gamma _d^p(C_m)= \infty \) , then \(\gamma _d^p(C_m \boxtimes C_n)=\infty \) .
