# covering a square with unit squares

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.)

• I meant to replace the original "any" with "some", which is done now. (Others may read the questioner's intent differently.) – Steve Huntsman Jun 25 '10 at 16:46
• Perhaps it should be stated for the record that this problem was originated by Alexander Soifer. – Joseph O'Rourke Jun 27 '10 at 11:25

Addendum in response to Ben Green's remark: I do have the 2008 Geombinatorics paper (but not the 2006 one). They define $\Pi(n)$ as the number of unit squares that can cover a square of side length $n+\epsilon$. It appears that the status as of this 2008 paper was that $\Pi(n)=n^2+O(n^{2/3})$ has been established, and they conjecture that $\Pi(n)=n^2+ \Omega(n^{1/2})$.
• Is there a proof that $\Pi(n) > n^2 + 1$? – Ben Green Jun 25 '10 at 19:22
• @Ben G.: Not that I can see, but then I have not scoured all this literature. $\Pi(1)=3$, but already for $n=2$ and $n=3$, the known lower bound is $n^2+1$. – Joseph O'Rourke Jun 25 '10 at 20:09