Triangles Covering Squares

The following pictures show n unit equilateral triangles covering the largest known square of side s.

1. 2. 3.
s = 2√3 - 3= .464+
Trivial. s = √3 - 1 = .732+
Found by David Cantrell
in August 2002. s = 4√3 - 6 = .928+
Found by David Cantrell
in August 2002.

4. 5. 6.
s = (3√3 - 3) / 2 = 1.098+
Found by Erich Friedman
in August 2002. s = (15√3 - 21) / 4 = 1.245+
Found by David Cantrell
in July 2005. s = (107√3 - 85) / 71 = 1.413+
Found by Maurizio Morandi
in April 2009.

7. 8. 9.
s = (7√3 - 6) / 4 = 1.531+
Found by Maurizio Morandi
in April 2009. s = (12 - 5√3) / 2 = 1.669+
Found by Maurizio Morandi
in April 2009. s = (139 - 69√3) / 11 = 1.771+
Found by Maurizio Morandi
in April 2009.

10. 11. 12.
s = 5 / √3 - 1 = 1.886+
Found by Maurizio Morandi
in April 2009. s = √3 + 1/4 = 1.982+
Found by Maurizio Morandi
in April 2009. s = (7√3 + 15) / 13 = 2.086+
Found by Maurizio Morandi
in April 2009.

1.		2.		3.
s = 2√3 - 3= .464+ Trivial.		s = √3 - 1 = .732+ Found by David Cantrell in August 2002.		s = 4√3 - 6 = .928+ Found by David Cantrell in August 2002.

4.		5.		6.
s = (3√3 - 3) / 2 = 1.098+ Found by Erich Friedman in August 2002.		s = (15√3 - 21) / 4 = 1.245+ Found by David Cantrell in July 2005.		s = (107√3 - 85) / 71 = 1.413+ Found by Maurizio Morandi in April 2009.

7.		8.		9.
s = (7√3 - 6) / 4 = 1.531+ Found by Maurizio Morandi in April 2009.		s = (12 - 5√3) / 2 = 1.669+ Found by Maurizio Morandi in April 2009.		s = (139 - 69√3) / 11 = 1.771+ Found by Maurizio Morandi in April 2009.

10.		11.		12.
s = 5 / √3 - 1 = 1.886+ Found by Maurizio Morandi in April 2009.		s = √3 + 1/4 = 1.982+ Found by Maurizio Morandi in April 2009.		s = (7√3 + 15) / 13 = 2.086+ Found by Maurizio Morandi in April 2009.