Paritetuille

Paritetuille is a hypothetical tiling concept in combinatorics and tiling theory. It denotes a tiling of a rectangular region by unit square tiles in which each tile carries a binary parity label, 0 or 1, and the labels satisfy a local parity constraint: the sum of the labels in every 2x2 block is even (mod 2). The term blends parity and tile and is used to study parity-driven structure in tilings.

Formally, for an m-by-n grid, a paritetuille is an assignment b_{i,j} ∈ {0,1} to each cell such that for all i = 1..m−1 and j = 1..n−1, b_{i,j} ⊕ b_{i+1,j} ⊕ b_{i,j+1} ⊕ b_{i+1,j+1} = 0. It does not change the placement of tiles (which are unit squares), but imposes a parity labeling. The number of distinct paritetuille labelings on an m×n region equals 2^{m+n−1}. Equivalently, any labeling can be written as b_{i,j} = r_i ⊕ c_j for some binary sequences r_1..r_m and c_1..c_n, with the representation invariant under flipping all r_i and all c_j simultaneously.

Variants may replace 2x2 blocks with k×k blocks, use different lattices (triangular, hexagonal), or permit additional constraints. The concept naturally leads to a linear-algebraic viewpoint over GF(2) and illustrates how local parity rules produce global structure.

Examples and applications: on a 2x2 region, there are 2^{3}=8 labelings; on a 3x4 region, 2^{6}=64. Paritetuille can inform theoretical explorations of parity constraints, puzzle design, and coding-theory analogies.

See also: tiling theory, parity, domino tilings, GF(2) linear systems.