Tile the unit square by, possibly infinitely many, squares of varying sizes, with at most three squares meeting at any corner. Color each square black or white independently, each with probability 1/2. Benjamini's square-tiling crossing conjecture. There is a constant c>0 such that the probability of a black left-to-right crossing is at least c.

Benjamini's square-tiling crossing conjecture

The conjecture asks whether a uniformly positive chance of a black left-to-right crossing exists for every square tiling of the unit square, even with infinitely many squares and black probability $1/2$ . No proposer or original date is identified in the retrieved sources.

Known results

A paper on site percolation in square packings proves a weaker statement for finite square packings when the white-coloring probability is an absolute constant $p$ sufficiently close to $1$ ; it does not establish the conjecture at $p=1/2$ or for infinite tilings. The source records the conjecture as open even at $p=2/3$ and, to the author's knowledge, for every fixed $p\in[1/2,1)$ .

Current status (as of August 2026): The stated conjecture at black probability $1/2$ , including possibly infinite tilings, remains open; only the restricted finite-tiling, highly biased-coloring result is recorded.

Benjamini's square-tiling crossing conjecture

Primary source

Progress summary

Solutions 0