A Unique Arithmetic Labeling of Hexagonal Lattices

(PREVIOUS TITLE: A UNIQUE ARITHMETIC LABELING OF MACROHEXAGONS) An r-layer macrohexagon has 3r sup 2 + 3r + 1 hexagonal cells. Can one label the cells by the set N sub r = {0,1,...3r sup 2 + 3r} such that each line of adjacent cells are labeled by numbers forming an arithmetic progression modulo (3r sup 2 + 3r + 1) (in proper order)? We show that for each r there exists such a labeling unique up to equivalence.