2013 USAMO 第 3 题

Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.

令 $n$ 为正整数。有 $\tfrac{n(n+1)}{2}$ 个标记，每个标记都有黑边和白边，排列成等边三角形，最大的一行包含 $n$ 个标记。最初，每个标记的黑色面朝上。操作是选择一条与三角形各边平行的线，然后翻转该线上的所有标记。如果可以通过执行有限数量的操作从初始配置获得该配置，则该配置被称为可接受的。对于每个允许的配置 $C$，让 $f(C)$ 表示从初始配置获得 $C$ 所需的最小操作数。求 $f(C)$ 的最大值，其中 $C$ 在所有允许的配置中变化。