Triangle Tiling III: The Triquadratic Tilings

An N-tiling of triangle ABC by triangle T (the “tile”) is a way of writing ABC as a union of N copies of T overlapping only at their boundaries. We continue the work of [1, 2] on classifying the possible triangle tilings. The results of all three papers together enable us to make some progress on two problems of Erdős about triangle tilings, one of which asks, “for which N is there an N-tiling of some triangle by some tile?”. This paper takes up a case that was left unsolved in [1, 2], which is the following. Let the tile T have angles α, β, and γ, and suppose 3α+ 2β = π, and α is not a rational multiple of π. Suppose there is an N-tiling of ABC. Then we classify the triples (ABC), T, N) such that there is an N-tiling of ABC by T . Our solution is as follows: There is an N-tiling of ABC by T if and only if N is a square times a product of distinct primes of the form 8n± 1, or 2, and the “tiling equation ” M +N = 2K is solvable in positive integers M and K with M < N and K divides M. In that case the tile can be taken to have sides a, b, and c where a = M and c = K and b = K −M/K, so all three sides will be integers. Combining this with the results of [2], we obtain a no-tiling theorem: if N is squarefree and N > 6 and divisible by at least one prime congruent to 3 mod 4, then there are no N-tilings of any triangle by any tile, unless the tile has a 120◦ angle.