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Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible total perimeter of the squares. Is $f(k^2+1)=4k$?
In [Er94b] Erdős dates this conjecture to 'more than 60 years ago'.

It is trivial from the Cauchy-Schwarz inequality that $f(k^2)=4k$. Erdős also asks for which $n$ is it true that $f(n+1)=f(n)$.