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HMMT 二月 2005 · TEAM2 赛 · 第 14 题

HMMT February 2005 — TEAM2 Round — Problem 14

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [30] Suppose that S tiles the set of all integer cubes. Prove that S has only one element.
解析
  1. [30] Suppose that S tiles the set of all integer cubes. Prove that S has only one element. Solution: Let the difference between the smallest and largest element of S be a . Then 3 3 the set equivalent to S that contains b can only contain integers between b − a and 3 3 b + a , inclusive. But for sufficiently large b , b is the only cube in this range, so S can only have one element.