Irredundant Covers of Square Grids by 2 x 2 Cards: A Defect Framework

Abstract: We study an irredundant covering of a square grid such that the cards can overlap and every card must lie inside the grid; every square of the grid must be covered and every card must have at least one square that is only covered by itself. We develop a generalized framework for the irredundant covering of a square grid with 2 by 2 cards. We first introduce a way of partitioning the cards to disjoint sets and using inclusion exclusion and other combinatorial arguments we derive a loose upper bound on the maximum size of an irredundant cover of the grid. After this loose bound we develop a framework to partition the grid squares in disjoint classes and using double counting and other methods we derive a tighter upper bound. Then we develop a uniform counting framework and a strict upper bound using boundary forcing, local overlap and defect arguments. We prove this bound to be a strict asymptotic bound by showing a lower bound that has the same leading coefficient. The 10 by 10 grid is treated as motivation and as a benchmark, not as a novelty claim; the novelty claim is the generalized framework for the irredundant covering.