HMMT 二月 2005 · TEAM1 赛 · 第 6 题

英文解析

6. [40] Show that for b even, there exists some M such that for every m, n > M with mneven, an m × n rectangle is (1 , b )-tileable.
   Solution: By the diagram below, it is possible to tile a (2 b + 2) × (4 b + 1) rectangle.
   Since we can already tile a (2 b + 2) × 2 b rectangle by above, and 2 b is relatively primeto 4 b + 1, this will allow us to tile any (2 b + 2) × n rectangle for n sufficiently large.
   Combining this with the previous problem, this will allow us to tile any m × n rectanglefor m and n sufficiently large and m even, completing the proof.
   To tile the (2 b + 2) × (4 b + 1) rectangle, we first tile the following piece:
   2 b-1 2 b 22
   b+1
   This is then combined with two 2 × 2 b rectangles, a 2 b × b rectangle, and a 2 b × (2 b + 1)
   rectangle as follows:
   2 x 2 b
   2 b x 2 b+1
   2 b x b
   2 x 2 b