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 = 2.558+
Found by Dominik Kamp
in April 2026.

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 = 2.82989+
Found by Jake Loyd
in June 2026.

16.
           17.
           18.
s = 2.9000+
Found by Ian Watson
in April 2026. 		s = 2.982+
Found by Maurizio Morandi
in July 2008. 		s = 3.051+
Found by Maurizio Morandi
in July 2008.

19.
           20.
           21.
s = 3.12929+
Found by Jake Loyd
in June 2026. 		s = 3.2305+
Found by Ian Watson
in April 2026. 		s = 3.31102+
Found by Thomas Schadt
in June 2026.

22.

s = 3.3757+
Found by Ian Watson
in April 2026. 		           23.

s = 3.4311+
Found by Ian Watson
in April 2026. 		           24.

s = 3.46780+
Found by Thomas Schadt
in June 2026.

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 = 3.6726+
Found by Ian Watson
in April 2026.

28.

s = 3.75707+
Found by Jake Loyd
in June 2026. 		           29.

s = 3.81711+
Found by Jake Loyd
in June 2026. 		           30.

s = 3.86619+
Found by Jake Loyd
in June 2026.

31.

s = 3.93107+
Found by Jake Loyd
in June 2026. 		           32.

s = 3.97880+
Found by Jake Loyd
in June 2026. 		           33.

s = 4.05164+
Found by Jake Loyd
in June 2026.

34.

s = 4.09088+
Found by Jake Loyd
in June 2026. 		           35.

s = 4.14123+
Found by Jake Loyd
in June 2026. 		           36.

s = 4.20791+
Found by Jake Loyd
in June 2026.

37.

s = 4.26880+
Found by Jake Loyd
in June 2026. 		           38.

s = 4.32018+
Found by Jake Loyd
in June 2026. 		           39.

s = 4.36792+
Found by Jake Loyd
in June 2026.

40.

s = 4.41058+
Found by Jake Loyd
in June 2026. 		           41.

s = 4.46446+
Found by Jake Loyd
in June 2026. 		           42.

s = 4.51177+
Found by Thomas Schadt
in June 2026.

43.

s = 4.55035+
Found by Thomas Schadt
in June 2026. 		           44.

s = 4.60023+
Found by Thomas Schadt
in June 2026. 		           45.

s = 4.67749+
Found by Thomas Schadt
in June 2026.

46.

s = 4.72965+
Found by Thomas Schadt
in June 2026. 		           47.

s = 4.78204+
Found by Thomas Schadt
in June 2026. 		           48.

s = 4.82632+
Found by Thomas Schadt
in June 2026.