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.)
[Math] covering a square with unit squares
coveringdiscrete geometrymg.metric-geometry
Related Solutions
There's a fair bit of literature about efficient packing of squares in squares. See, e.g., Kearney and Shiu, Efficient packing of unit squares in a square, available at http://www.emis.de/journals/EJC/Volume_9/PDF/v9i1r14.pdf, also Walter Stromquist, Packing 10 or 11 unit squares in a square, available at http://www.combinatorics.org/Volume_10/PDF/v10i1r8.pdf
Four pieces, using the tessellation technique I learned from Harry Lindgren's Geometric Dissections (1964):
Best Answer
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.