# Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few days, but so far I'm not even sure what's the best way to formalize it. Has this problem been solved? Is there a useful hint? Can you recommend me something to read on the subject?

UPD: my best idea so far is to represent a rectangle in $\mathbb{E}^2$ by a tuple $r = (p, x, y)$, where $p$ is a point, and $x$ and $y$ are two orthogonal vectors representing two sides coming out of it. Then we define symmetries as (a) euclidean motions acting as usual, and (b) swapping of $x$ and $y$. Then we observe that the solutions of the problem, that is, $m$ touching rectangles, are mapped to some other solutions under euclidean motions and dilatations of the underlying space, as well as under $S_m$. Now we have to algebrize the problem, but I'm unsure how.

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Three is easy, and $K_5$ is known not to be a planar graph, so it comes down to: can you do four? Size seems to be a killer. Three touching rectangles form a kind of a closed path that must either go around the fourth or have the fourth wrap around somehow. Doesn't seem to work, but I may have missed something. –  Jyrki Lahtonen Jul 9 '11 at 6:04
Four is easy (three in a stack, and another one on the side). –  Alexei Averchenko Jul 9 '11 at 6:10
Are top and bottom of the stack touching? –  Jyrki Lahtonen Jul 9 '11 at 6:15
\$Jurki: Oh..... –  Alexei Averchenko Jul 9 '11 at 6:18
Removed the (hint) tag. See here. I think it is sufficient to just ask for a hint, like you did in the question. No need to make a tag out of it. –  Willie Wong Jul 10 '11 at 11:47