Conway Unfoldings of Level 1 Menger Polycubes

We give an explicit affirmative answer to Langerman and Winslow’s question about Conway unfoldings of the level-1 Menger sponge. Its 72 exposed unit squares develop into 36 columns of height two, with retained hinges forming a path through all surface faces. The footprint has perimeter 114 and tiles the plane by translations. Cuts follow unit-cube edges, no face is subdivided, and coincident cut edges are allowed. A second construction covers every level-1 (1,1,n)-Menger polycube, of genus 4n+1, by joining hinged dominoes into a chain. The contribution is a Conway unfolding for this specific Menger family. We explain the geometric search restriction and prove the family construction from finite periodic tables, printed in an appendix. A small companion package supplies exact corner maps, hinge lists, templates and a standalone integer verifier.