Triangles in Squares

The following pictures show n equilateral triangles with side 1 packed inside the smallest known square (of side length s).


1.
           2.
           3.
s = (√2 + √6) / 4 = .965+
Trivial.
s = √6/2 = 1.224+
Trivial.
s = √3/2 + √6/4 = 1.478+
Found by Erich Friedman
in 1996.


4.
           5.
           6.
s = 1 + 1 / √3 = 1.577+
Found by Erich Friedman
in 1996.
s = 1.803+
Found by Erich Friedman
in 1996.
s = 9 / 2 - 3 √3 / 2 = 1.901+
Found by Erich Friedman
in 1996.


7.
           8.
           9.
s = 2
Found by Erich Friedman
in 1996.
s = 3 √3 / 2 - 1 / 2 = 2.098+
Found by Erich Friedman
in 1996.
s = 2.287+
Found by Maurizio Morandi
in July 2008.


10.
           11.
           12.
s = 2.377+
Found by David W. Cantrell
in July 2002.
s = 2.490+
Found by Maurizio Morandi
in June 2008.
s = 7√3/2 - 7/2 = 2.562+
Found by David W. Cantrell
in July 2002.


13.
           14.
           15.
s = 2.595+
Found by David W. Cantrell
in July 2002.
s = 2.726+
Found by David W. Cantrell
in July 2002.
s = 2.830+
Found by Maurizio Morandi
in July 2008.


16.
           17.
           18.
s = 2.900+
Found by Maurizio Morandi
in July 2008.
s = 2.982+
Found by Maurizio Morandi
in July 2008.
s = 3.051+
Found by Maurizio Morandi
in July 2008.


19.
           20.
           21.
s = 4 - √3 / 2 = 3.133+
Found by Maurizio Morandi
in July 2008.
s = 3.233+
Found by Károly Hajba
in August 2008.
s = 10√3 - 14 = 3.320+
Found by Maurizio Morandi
in July 2008.


22.

s = 3.378+
Found by David W. Cantrell
in August 2012.
           23.

s = 3.436+
Found by Maurizio Morandi
in August 2008.
           24.

s = 3.468+
Found by Maurizio Morandi
in August 2008.


25.

s = 3.537+
Found by David W. Cantrell
in August 2012.
           26.

s = 3.575+
Found by Maurizio Morandi
in September 2008.
           27.

s = 3.675+
Found by Maurizio Morandi
in September 2008.


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