The \(2\)-Packing Number of \(3\)-Dimensional Grids

What is the 2-packing number of the \(1 \times m \times n\) complete grid graph? Fisher solved this for \(1 \times m \times n\) grids for all \(m\) and \(n\). We answer this for \(2 \times m \times n\) grids for all \(m\) and \(n\), and for \(3 \times 3 \times n\), \(3 \times 4 \times n\), \(3 \times 7 \times n\), \(4 \times 4 \times n\), and \(5 \times 5 \times n\) grids for all \(n\). Partial results are given for other sizes.