HMMT 二月 2005 · TEAM1 赛 · 第 5 题
HMMT February 2005 — TEAM1 Round — Problem 5
题目详情
- [25] Show that for b even, there exists some M such that for every n > M , a 2 b × n rectangle is (1 , b )-tileable.
解析
- [25] Show that for b even, there exists some M such that for every n > M , a 2 b × n rectangle is (1 , b )-tileable. Solution: Recall from above that we can tile a 2 × 2 b rectangle. Four columns of a b ( b + 1) × 2 b rectangle can be tiled as shown below, and repeating this times tiles 2 the entire rectangle. Since any integer at least b can be written as a positive linear combination of 2 and b + 1, we can tile any 2 b × n rectangle for n ≥ b .