Iwan Praton

  1. Tiling a unit square with 8 squares.

    Authors: Iwan Praton
    Subjects: Metric Geometry
    Abstract

    Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote
    the maximum values of the sum of the edge lengths of the n small squares, where
    in the case of f(n) the maximum is taken over all arbitrary packings of the
    unit square, and in the case of g(n) it is taken over all tilings of the unit
    square (i.e., the total area of the n small squares is 1). Benton and Tyler
    asked for which values of n we have f(n)=g(n). We show that f(8)>g(8). More
    precisely, we show that g(8)=13/5; it is known that f(8) is at least 8/3.

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