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HMMT 二月 2005 · TEAM1 赛 · 第 1 题

HMMT February 2005 — TEAM1 Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [15] Prove that for any two types of dominoes, there exists a rectangle that can be tiled by dominoes of either type.
解析
  1. [15] Prove that for any two types of dominoes, there exists a rectangle that can be tiled by dominoes of either type. Solution: Note that a type ( a, b ) domino tiles a max { 1 , 2 a }× 2 b rectangle (see diagram ′ ′ for a > 0). Then both type ( a, b ) and type ( a , b ) dominoes tile a (max { 1 , 2 a } · ′ ′ max { 1 , 2 a } ) × (2 b · 2 b ) rectangle. b b a a