A Complete Combinatorial Classification of a Finite Residue Covering Problem on [34, 66]
Jonathan Michael Langford, [email protected] · Zenodo (CERN European Organization for Nuclear Research) · 2026
We present a complete combinatorial classification of a finite residue-covering problem on the interval $I=[34,66]\cap\mathbb{Z}$. The problem asks for the maximum number of integers in I that can be covered by selecting one residue class modulo 2 and two distinct residue classes modulo each of 3, 5, and 7. While exhaustive search over the 1,260 canonical assignments—equivalently, 10,080 ordered assignments—gives maximum coverage 32, with 126 canonical optima and 1,008 ordered optima, we derive these conclusions combinatorially. By mapping the interval to a punctured $5\times7$ toroidal grid via the Chinese Remainder Theorem, we reduce the problem to a mod-6 grid-coloring puzzle. We prove the Singleton-Class Theorem, which equates optimal assignments to singleton colors on residual subgrids. By explicitly counting singleton incidences and analyzing color-pair compatibility, we analytically derive the seed distribution (92, 110, 8), proving the exact counts of optimal assignments and the strict maximum coverage bound.