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  • $\begingroup$ So... A248183 suggested $\sqrt{2}/2$ asymptotic... which is really close to the upper bound $\sqrt{3}-1$. This suggested some mixed pattern though $\endgroup$
    – JetfiRex
    Commented Jan 22, 2022 at 17:41
  • $\begingroup$ Good news: That friend of mine suggested a $\sqrt 2/2$ upper bound, which, If your hypothesis (the exact number somewhat matches the sequence you gave me) is correct and we can construct using the pattern, this upper bound and lower bound matches. $\endgroup$
    – JetfiRex
    Commented Jan 22, 2022 at 22:29
  • $\begingroup$ I'm sorry, but can I ask you to do me a favor... It seems that the construction has a central square boundary and there seem to be vague boundary on the four corners... Just like the location of the green entries in the proof of $n/\sqrt 2$. If you can give an asymptotic construction of $n/\sqrt 2$ based on your findings, I will really appreciate you for your work... Thank you so much for your work in advance! $\endgroup$
    – JetfiRex
    Commented Jan 23, 2022 at 1:57
  • $\begingroup$ Is there a solution with $D_4$ or maybe even $D_8$ symmetry? $\endgroup$
    – Sean Eberhard
    Commented Jan 23, 2022 at 10:16
  • 1
    $\begingroup$ @Wolfgang I named them Figure 1 through 11 just now. $\endgroup$
    – RobPratt
    Commented Jan 28, 2022 at 22:40