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HMMT 二月 2005 · TEAM2 赛 · 第 13 题

HMMT February 2005 — TEAM2 Round — Problem 13

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [25] Exhibit a set S which tiles the integers Z but not the natural numbers N .
解析
  1. [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, it tiles Z by translation by multiples of 4. On the other hand, it is easy to see that it cannot tile N : 1 is contained in { 1 , 3 , 4 , 6 } , but then there is no possible set for 2 to be contained in that does not overlap. Another example is { 1 , 2 , 6 } .