Squares in Circles

The following pictures show n unit squares packed inside the smallest known circle (of radius r).
Only the cases n=1 and n=2 have been proved optimal.


1.            2.            3.
r = √2 / 2 = .707+
Trivial.
r = √5 / 2 = 1.118+
Trivial.
r = 5 √17 / 16 = 1.288+
Found by Erich Friedman
in 1997.


4.            5.            6.
r = √2 = 1.414+
Found by Erich Friedman
in 1997.
r = √10 / 2 = 1.581+
Found by Erich Friedman
in 1997.
r = 1.688+
Found by Erich Friedman
in 1997.


7.            8.            9.
r = √13 / 2 = 1.802+
Found by Erich Friedman
in 1997.
r = 1.978+
Found by David W. Cantrell
in March 2002.
r = √1105 / 16 = 2.077+
Found by Erich Friedman
in 1997.


10.            11.            12.
r = 3 √2 / 2 = 2.121+
Found by Erich Friedman
in 1997.
r = 2.214+
Found by David W. Cantrell
in March 2002.
r = √5 = 2.236+
Found by Erich Friedman
in 1997.


13.            14.            15.
r = 2.3607+
Found by David W. Cantrell
in November 2010.
r = 2.5
Found by Erich Friedman
in 1997.
r = 2.533+
Found by by David W. Cantrell
in June 2002.


16.            17.            18.
r = √11009 / 40 = 2.623+
Found by Erich Friedman
in 1997.
r = 2.677+
Found by Maurizio Morandi
in December 2010.
r = √481 / 8 = 2.741+
Found by David W. Cantrell
in March 2002.


19.            20.            21.
r = 2.807+
Found by David W. Cantrell
in June 2002.
r = 2.893+
Found by David W. Cantrell
in June 2002.
r = √34 / 2 = 2.915+
Found by Erich Friedman
in 1997.


22.            23.            24.
r = 3.036+
Found by David W. Cantrell
in June 2002.
r = 3.073+
Found by David W. Cantrell
in June 2002.
r = 3.110+
Found by Maurizio Morandi
in October 2010.


25.            26.            27.
r = √5809 / 24 = 3.175+
Found by David W. Cantrell
in April 2002.
r = √41 / 2 = 3.201+
Found by Erich Friedman
in 1997.
r = 13√145/48 = 3.261+
Found by Maurizio Morandi
in September 2010.


28.            29.            30.
r = 3.3471+
Found by David W. Cantrell
in November 2010.
r = 3.412+
Found by DC after MM
in August 2012.
r = √27985 / 48 = 3.485+
Found by David W. Cantrell
in March 2002.


31.            32.            33.
r = 5 √2 / 2 = 3.535+
Found by Erich Friedman
in 1997.
r = √40385 / 56 = 3.588+
Found by Erich Friedman
in 1997.
r = √13 = 3.605+
Found by Maurizio Morandi
in June 2010.


34.            35.
r = 3.629+
Found by Maurizio Morandi
in December 2010.
r = √53 / 2 = 3.640+
Found by David Eppstein
in March 2002.


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