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  • $\begingroup$ Do you insist on the squares being aligned, that is, not rotated with respect to the big square? $\endgroup$
    – Gerry Myerson
    Commented Feb 20, 2021 at 21:48
  • $\begingroup$ @GerryMyerson Yes. Sorry if I'm not being very formal with the parameters of the question, but indeed the squares have to be sort of "snapped to" the integer coordinates of the plane. $\endgroup$
    – Marcel K. Goh
    Commented Feb 20, 2021 at 21:57
  • $\begingroup$ Trivial update (posted here to avoid spamming the homepage, hopefully): $f(11) = 35$. $\endgroup$
    – Marcel K. Goh
    Commented Feb 20, 2021 at 22:04
  • $\begingroup$ The $n$ used as side length and the $n$ used as number of smaller squares don't seem to be the same in your examples. Is that right? If so you should use different letters for these two numbers. $\endgroup$
    – Sam Hopkins
    Commented Feb 20, 2021 at 23:28
  • $\begingroup$ @SamHopkins They are supposed to be the same $n$. The peculiar case is $n=8$, when only $7$ squares are needed to produce a maximal sum. Of course, we can artificially decompose one of the squares into two smaller ones, giving us $8$ squares, as prescribed. $\endgroup$
    – Marcel K. Goh
    Commented Feb 20, 2021 at 23:42