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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

It is easy to do four if they are allowed to touch at a point - a windmill configuration will work. There is an easy argument about meeting along line segments - consider the orientation of the rectangles, and once you have two think about where a third would have to go.

If they have to touch along non-empty line segments, then it is easy to see that they have to be orthogonal to each other - each rectangle has to be in one of two orientations. Put two rectangles together. There is a site at each end of the line segment where they meet where a third rectangle can be placed to touch both - two sites. To fit four rectangles you'd have to use both sites, but then the two added rectangles can't touch. [consider the limited number of cases if necessary] Also note that each pair of rectangles, once you have three, defines its own site for the next rectangle (just one, because the other one is the point where all three meet).

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