Can some square of side length greater than $n$ be covered by $n^2+1$ unit squares? (The unit squares may be rotated. The large square and its interior must be covered.)



covering a square with unit squares
This reference is certainly pertinent, being the second Google hit for "covering a square with squares" (after your question). Just reading it now...


<http://www.uccs.edu/~faculty/asoifer/docs/untitled.pdf>


UPDATE: so far as I can tell from looking at this article, the author regards your question as an unsolved problem. There is a further article by him and Karabash, apparently in (his) journal Geombinatorics, vol. 18, which I cannot access online and which has not been reviewed on MathSciNet.


