Minimal and maximal Numbrix puzzles

This paper explores special arrangements of clues in $m \times n$ Numbrix puzzles. The maximum number of clues which fails to define an $m \times n$ puzzle is demonstrated for all $m$ and $n$. In addition, a small upper bound on the minimum number of clues required to define an $m \times n$ puzzle is given for all $m$ and $n$ as well. For small $m \geq 3$ our upper bound appears to actually give the minimum number and hence we conjecture that our bound may be sharp for all $m \geq 3$.