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An explicit counterexample to the six-neighbor square tiling conjecture

Primarily human-written textHuman understanding: all partscs.DM — Discrete Mathematics

Contributed by Chunlei Liu ↗, Aijin Lin

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

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 construction is locally finite and gives a counterexample on the whole plane.

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Version history

  1. v1Submitted by Chunlei LiuInitial depositCurrentOct 19, 2026