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 = 6√3 - 9 = 1.392+
Found by Erich Friedman
in August 2002.

7. 8. 9.
s = (11√3 - 16) / 2 = 1.526+
Found by David Cantrell
in July 2005. s = 5√3 - 7 = 1.660+
Found by David Cantrell
in July 2005. s = (13√3 - 19) / 2 = 1.758+
Found by David Cantrell
in July 2005.

10. 11. 12.
s = 8√3 - 12 = 1.856+
Found by David Cantrell
in July 2005. s = (43√3 - 49) / 13 = 1.959+
Found by David Cantrell
in July 2005. s = (50√3 - 60) / 13 = 2.046+
Found by David Cantrell
in July 2005.

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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 = 6√3 - 9 = 1.392+ Found by Erich Friedman in August 2002.

7.		8.		9.
s = (11√3 - 16) / 2 = 1.526+ Found by David Cantrell in July 2005.		s = 5√3 - 7 = 1.660+ Found by David Cantrell in July 2005.		s = (13√3 - 19) / 2 = 1.758+ Found by David Cantrell in July 2005.

10.		11.		12.
s = 8√3 - 12 = 1.856+ Found by David Cantrell in July 2005.		s = (43√3 - 49) / 13 = 1.959+ Found by David Cantrell in July 2005.		s = (50√3 - 60) / 13 = 2.046+ Found by David Cantrell in July 2005.