{"id":"hexagon/2610.00394","pdf":"https://hexagonmath.org/pdf/2610.00394v1","url":"https://hexagonmath.org/abs/2610.00394v1","tags":["msc:52C20","msc:52C26","msc:05B45","Square Tiling Conjecture"],"score":0,"title":"An explicit counterexample to the six-neighbor square tiling conjecture","domain":"cs.DM","source":"hexagon","authors":["Chunlei Liu","Aijin Lin"],"reviews":0,"subject":"cs.DM","summary":"Luo, Sun and Wu conjectured that in a locally finite square tiling of the complex plane in which each square intersects exactly six others, all squares have the same size. We construct a counterexample by surrounding a ten-square core with alternating rectangular shells, each formed by six squares. The only point-only contacts occur at the three four-square points in the core; all subsequent contacts have positive length. A two-state induction shows that every new square has four contacts when inserted and acquires two more at the next shell. The nested rectangles exhaust the plane, so the…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}