I have some box with dimensions x,y,z. I put a net around it which includes the top and bottom. The net has unit squares on it. Whats the maximum amount of cuts you can make on the net but still have it in one piece?
You would want to make enough cuts to reduce the net to a spanning tree. The number of edges in a tree is one less than the number of vertices. So, here's what you do: find the number of vertices, $v$; find the number of edges, $e$, in the uncut net; then you can make $e-v+1$ cuts, leaving $v-1$ edges uncut. Can you work out $v$ and $e$ from $x,y,z$?