Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose I have a rectangle $R$ tiled by finitely many squares - i.e. written as a finite almost disjoint union $$R = \bigcup_{i=1}^n S_i $$ where each $S_i$ is a square and such that $S_i^\circ \cap S_j^\circ = \varnothing$ whenever $i \neq j$ (so that the squares can only overlap at their edges).

Is it always possible to write $R$ as an (almost disjoint) union $$R = R_1 \cup R_2,$$ where $R_1$ and $R_2$ are rectangles formed from the squares $S_i$? See the badly drawn image below for an example (using your imagination to turn the wonky rectangles into squares)

$\hskip1.5in$ enter image description here

(This question was inspired by Can a rectangle be written as a finite almost disjoint union of squares? - it follows from the discussion there that the side lengths of R are necessarily commensurable. I have no idea what the answer to this question is - it seems like something that should be false, but I can't come up with a counterexample!)

share|cite|improve this question
up vote 1 down vote accepted

Not necessarily. There are many examples. You can find one at this Wikipedia link.

share|cite|improve this answer
Easy! Thanks for that. Now I can stop thinking about it – Alex Amenta Apr 28 '12 at 4:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.