HMMT 二月 2005 · TEAM2 赛 · 第 13 题

英文解析

13. [25] Exhibit a set S which tiles the integers Z but not the natural numbers N .
    Solution: One example is { 1 , 3 , 4 , 6 } . Since its elements are all distinct modulo 4, ittiles Z by translation by multiples of 4. On the other hand, it is easy to see that itcannot tile N : 1 is contained in { 1 , 3 , 4 , 6 } , but then there is no possible set for 2 tobe contained in that does not overlap. Another example is { 1 , 2 , 6 } .