Using squares of size $1$ to $n$, make a patch that can tile the plane. Solutions for $n=2, 3, 4, 5$ are easy to find.
Example for 6:
Example for 7:
Example for 8:
Example for 9:
I haven't found a solution for 10 yet. This one lacks the unit square:
Is there a shape made from squares of size $1$ to $10$ that can tile the plane? Are there solutions for higher $n$? For $n$ that admit solutions, how many basic solutions are there?