TITLE: Tiling the plane with consecutive squares
QUESTION [18 upvotes]: For which $n$ is it possible to find a region $R$ made of non-overlapping squares of side length $1,2,\ldots,n$ which tiles the plane?
$n=1$ is trivial, and $n=2$ works as well. However, for $n\geq3,$ I am unable to find $R$ that work. Obviously, we can try every possible combination for smaller values, but I want to know for arbitrary $n$ what the conditions are for $R$ to exist.

REPLY [2 votes]: Here's another way of constructing these. I start with the area of the first N squares, eg for 1 to 9 the area is 285 which is 3x5x19. Then I try to tile a torus of 15x19, then klein bottles of 15x19 and 19x15, finally a manifold (real projective plane) of those dimensions. This all fails for 9, so I use two sets and try torus, klein bottles, rpp for whichever rectangle sizes are convenient. Sooner or later one will work. Here are two sets for 1 to 9 in a 19x30 klein bottle, doubled and flipped so it forms a patch which tiles the plane by translation only. The resulting four sets of squares 1 to 9 are coloured differently.

For 1 to 10 I had to use three sets on a 33x35 torus so I get a translation-only patch with three sets. I told the tiling program to make sure each of the three sets was linked into one self touching area, but if such a tiling did not exist we could use disjoint sets too.

It seems that when we allow multiple sets like this, the probability of a tiling increases with N but a proof is not in my skill set. I might try a few larger N to see whether my program can manage the tiling in reasonable time. One thing I noticed is that the area always seems to be nicely composite, rather than a prime or near prime.
Here is another triple set for 1 to 11, on a cylinder. Note that the sets are contiguous, blue and orange join when wrapped top and bottom.

And here's a two-set tilings for 1 to 11 using a 44x23 Klein bottle. This took my program 86 hours to find.

Just found a 1 to 15 with two sets on a cylinder.

Here's a triple 1 to 12 on a toroid