HMMT 二月 2005 · TEAM1 赛 · 第 12 题

英文解析

12. [20] Suppose the elements of A are either bounded below or bounded above. Showthat if S tiles A , then it does so uniquely, i.e., there is a unique tiling of A by S .
    Solution: Assume A is bounded below; the other case is analogous. In choosingthe tiling of A , note that there is a unique choice for the set S that contains the minimum element of A . But then there is a unique choice for the set S that contains 0
    the minimum element of A \ S . Continuing in this manner, there is a unique choice 1
    for the set containing the minimum element not yet covered, so we see that the tiling 0
    is uniquely determined.