Conway criterion

The Conway criterion is a sufficient condition for a polyomino to tile the plane. It tests whether the boundary can be divided into matching segments that pair under translations or rotations.

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Polyomino: A plane figure formed by joining congruent squares edge to edge. The criterion applies to polyominoes and examines the arrangement of their square edges.
Tiling: A covering of a region by shapes whose interiors do not overlap. The criterion establishes a way for repeated copies to cover the entire plane.
Necessary condition: A condition that must hold whenever a specified conclusion is true. Failure of the Conway criterion does not establish that a polyomino cannot tile.
Domino tiling: A tiling by dominoes, each formed from two edge-adjacent unit squares. The domino is a basic polyomino whose repeated copies tile the plane.
Boundary word: A symbolic encoding of an oriented polygon boundary as a sequence of edge directions and labels. The criterion’s boundary segments are represented and compared through directional sequences.
Square lattice: The grid of points with integer coordinates in the Euclidean plane. Polyomino edges follow the horizontal and vertical directions of this grid.
Coloring argument: A proof technique that assigns colors to positions and tracks color counts under allowed placements. Colorings can prove non-tileability even when boundary pairing gives no answer.
L-tromino: A three-square polyomino shaped like the letter L. Its boundary can be checked directly against the criterion’s segment-pairing condition.
Translation tiling: A tiling formed by copies of a shape moved only by translations. Matching boundary segments support repeating copies through translational placement.
Grid graph: A graph whose vertices and edges form a regular square grid. It provides a graph representation of the unit squares and their shared edges.