Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Prove it United States

Problem:

Show that no rectangle of the form $1 \times k$ or $2 \times n$ , where $4 \nmid n$ , is $(1,2)$ -tileable.

Solution

Solution:

The claim is obvious for $1 \times k$ rectangles. For the others, color the first two columns black, the next two white, the next two black, etc. Each $(1,2)$ domino will contain one square of each color, so in order to be tileable, the rectangle must contain the same number of black and white squares. This is the case only when $4 \mid n$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.