HMMT 二月 2005 · TEAM1 赛 · 第 12 题

HMMT February 2005 — TEAM1 Round — Problem 12

[20] Suppose the elements of A are either bounded below or bounded above. Show that if S tiles A , then it does so uniquely, i.e., there is a unique tiling of A by S .

解析

[20] Suppose the elements of A are either bounded below or bounded above. Show that 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 choosing the tiling of A , note that there is a unique choice for the set S that contains the 0 minimum element of A . But then there is a unique choice for the set S that contains 1 the minimum element of A \ S . Continuing in this manner, there is a unique choice 0 for the set containing the minimum element not yet covered, so we see that the tiling is uniquely determined.