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.