Within the integer lattice \(\mathbb{Z}^d\), an important role -- somewhat akin to that of compact convex sets in Euclidean space -- is played by the rectangular blocks (or meshes). \textit{J. B. Baillon} and \textit{S. Simons} [J. Comb. Theory, Ser. A 60, No. 1, 147--154 (1992; Zbl 0764.05009)] proved an almost-fixed-point theorem and a fixed-point theorem for mappings of rectangular blocks into \(\mathbb{Z}^d\). These theorems are analogous, respectively, to the Halpern-Bergman fixed-point theorem and the Browder fixed-point theorem for multifunctions. While Baillon and Simons made significant use of those topological results in the proofs of their theorems, they speculated about the possibility of a direct combinatorial proof. This paper presents such proofs for both theorems. Moreover, it is shown that the Halpern-Bergman theorem follows quickly from the almost-fixed-point theorem, giving a sort of informal equivalence between the two as well as a formal similarity. The interested reader should probably also consult [\textit{M. B. Smyth} and \textit{R. Tsaur}, Appl. Categ. Struct. 11, No. 1, 95--116 (2003; Zbl 1023.54033)] for a more general introduction to this deep and interesting topic, which has potential applications to image processing, numerical analysis, and other areas.