HMMT 二月 2005 · TEAM1 赛 · 第 4 题

英文解析

4. [35] Show that all other rectangles of even area are (1 , 2)-tileable.
   Solution: First, we demonstrate that there exist (1 , 2)-tilings of 2 × 4, 3 × 4, 3 × 6,
   and 5 × 6 rectangles.
   Now, notice that by combining these rectangles, we can form any rectangle of evenarea other than those described in the previous problem: using the first rectangle, wecan form any 2 × n rectangle with 4 | n . By combining the first two, we can form anym × 4 rectangle with m ≥ 2, and by combining the last three, we can form any m × 6
   rectangle with m ≥ 3. From these, we can form any m × n rectangle with m ≥ 3 andn even and greater than 2, completing the proof.