2023/06/15 by Baek, Jineon, Lee, Seewoo
#52C15 (Primary) 05B40 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2306.09533
Conway and Soifer showed that an equilateral triangle T of side n + ε with sufficiently small ε > 0 can be covered by n2 + 2 unit equilateral triangles. They conjectured that it is impossible to cover T with n2 + 1 unit equilateral triangles no matter how small ε is. We show that if we require all sides of the unit equilateral triangles to be parallel to the sides of T (e.g. \bigtriangleup and \bigtriangledown), then it is impossible to cover T of side n + ε with n2 + 1 unit equilateral triangles for any ε > 0. As the coverings of T by Conway and Soifer only involve triangles with sides parallel to T, our result determines the exact minimum number n2+2 of unit equilateral triangles with all sides parallel to T that cover T. We also determine the largest value ε = 1/(n + 1) (resp. ε = 1 / n) of ε such that the equilateral triangle T of side n + ε can be covered by n2+2 (resp. n2 + 3) unit equilateral triangles with sides parallel to T, where the first case is achieved by the construction of Conway and Soifer.