HMMT 二月 2005 · TEAM2 赛 · 第 3 题
HMMT February 2005 — TEAM2 Round — Problem 3
题目详情
- [20] Prove that every ( a, b )-tileable rectangle contains a rectangle of these dimensions.
解析
- [20] Prove that every ( a, b )-tileable rectangle contains a rectangle of these dimensions. Solution: An m × n rectangle, m ≤ n , does not contain a max { 1 , 2 a } × 2 b rectangle if and only if m < 2 a or n < 2 b . But if either of these is the case, then the square closest to the center of the rectangle cannot be paired with any other square of the rectangle to form a domino of type ( a, b ), so the rectangle cannot be ( a, b )-tileable.