HMMT 二月 2005 · TEAM1 赛 · 第 3 题

英文解析

3. [10] Show that no rectangle of the form 1 × k or 2 × n , where 4 - n , is (1 , 2)-tileable.
   Solution: The claim is obvious for 1 × k rectangles. For the others, color the first twocolumns black, the next two white, the next two black, etc. Each (1 , 2) domino willcontain one square of each color, so in order to be tileable, the rectangle must containthe same number of black and white squares. This is the case only when 4 | n .