HMMT 二月 2003 · 冲刺赛 · 第 20 题

英文解析

20. What is the smallest number of regular hexagons of side length 1 needed to completelycover a disc of radius 1?
    Solution: 3
    First, we show that two hexagons do not suffice. Specifically, we claim that a hexagoncovers less than half of the disc’s boundary. First, a hexagon of side length 1 may beinscribed in a circle, and this covers just 6 points. Translating the hexagon verticallyupward (regardless of its orientation) will cause it to no longer touch any point onthe lower half of the circle, so that it now covers less than half of the boundary. Byrotational symmetry, the same argument applies to translation in any other direction,
    proving the claim. Then, two hexagons cannot possibly cover the disc.
    The disc can be covered by three hexagons as follows. Let P be the center of the circle.
    Put three non-overlapping hexagons together at point P . This will cover the circle,
    °
    since each hexagon will cover a 120 sector of the circle.