@MISC{Beeson10tilinga, author = {Michael Beeson}, title = {Tiling a Triangle with Congruent Triangles}, year = {2010} }

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Abstract

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), which may or may not be similar to ABC. We wish to characterize the numbers N for which some triangle ABC can be tiled by N tiles, or more generally to characterize the triples (N, T) such that ABC can be N-tiled using tile T. In the first part of the paper we exhibit certain families of tilings which contain all known tilings. We conjecture that the exhibited tilings are the only possible tilings. If that is so, then for there to exist an N-tiling of any triangle ABC, N must be a square, or 2, 3, or 6 times a square, or a sum of two squares. We were able to reduce this conjecture to a special case. The case we could not solve is when tile has angles α, β, and γ with 3α = 2β, and sin(α/2) is rational. Some number-theoretic properties of N are also necessary. The triangle ABC must have angles 2α, β, and β + γ and α is not a rational multiple of π. The simplest unsolved case is N = 28, with a tile whose sides are 2, 3, and 4, and triangle ABC has sides 12, 14, and 16. In particular, there are no N-tilings for N = 7. We have earlier given a (rather long) traditional Euclid-style proof of the impossibility of a 7-tiling, but could not handle even N = 11,