Triangles in Squares

The following pictures show n equilateral triangles packed inside a unit square, so that the total sum of the sides of the triangles is as large as possible.

1.
           2.
           3.
Σ s = √6–√2 = 1.03527+
Trivial.
  		Σ s = 2√6/3 = 1.63299+
Trivial.
A continuum of solutions. 		Σ s = 1+1/√3+√(2–√3) = 2.09498+
Found by Jonathan Viquerat
in June 2026.

4.
           5.
           6.
Σ s = 6–2√3 = 2.53589+
Found by Jonathan Viquerat
in June 2026. 		Σ s = (5+7√3)/6 = 2.85405+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 7/3+√3/2 = 3.19935+
Found by Jonathan Viquerat
in June 2026.

7.
           8.
           9.
Σ s = 5/2+7/4√3 = 3.51036+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5–2/√3 = 3.84529+
Found by Jonathan Viquerat
in June 2026. 		Σ s = (35√3–24)/9 = 4.06908+
Found by Jonathan Viquerat
in June 2026.

10.
           11.
           12.
Σ s = 4.27964+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 4.54529+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 4.77454+
Found by Jonathan Viquerat
in June 2026.

13.
           14.
           15.
Σ s = 5.03476+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5.27330+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5.42874+
Found by Jonathan Viquerat
in June 2026.

16.
           17.
           18.
Σ s = 5.65499+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 5.83554+
Found by Haowei Lin
in July 2026. 		Σ s = 6.03574+
Found by Jonathan Viquerat
in June 2026.

19.
           20.
           21.
Σ s = 6.21461+
Found by Jonathan Viquerat
in June 2026. 		Σ s = 6.36146+
Found by Haowei Lin
in July 2026. 		Σ s = 6.46072+
Found by Joost de Winter
in September 2026.

22.
           23.
           24.
Σ s = 6.64779+
Found by Joost de Winter
in October 2026. 		Σ s = 6.86401+
Found by Joost de Winter
in October 2026. 		Σ s = 7.02040+
Found by Joost de Winter
in October 2026.

25.
           26.
           27.
Σ s = 7.17519+
Found by Joost de Winter
in October 2026. 		Σ s = 7.36390+
Found by Joost de Winter
in September 2026. 		Σ s = 7.46549+
Found by Joost de Winter
in October 2026.

28.
           29.
           30.
Σ s = 7.62397+
Found by Joost de Winter
in September 2026. 		Σ s = 7.71948+
Found by Joost de Winter
in October 2026. 		Σ s = 7.84284+
Found by Joost de Winter
in September 2026.

31.
           32.
           33.
Σ s = 8.02396+
Found by Joost de Winter
in September 2026. 		Σ s = 8.11290+
Found by Joost de Winter
in October 2026. 		Σ s = 8.21782+
Found by Joost de Winter
in September 2026.

34.
           35.
           36.
Σ s = 8.39422+
Found by Joost de Winter
in September 2026. 		Σ s = 8.48960+
Found by Joost de Winter
in October 2026. 		Σ s = 8.65967+
Found by Joost de Winter
in September 2026.

37.
           38.
           39.
Σ s = 8.74553+
Found by Joost de Winter
in October 2026. 		Σ s = 8.90856+
Found by Joost de Winter
in October 2026. 		Σ s = 9.05475+
Found by Joost de Winter
in October 2026.

40.
           41.
           42.
Σ s = 9.20058+
Found by Joost de Winter
in September 2026. 		Σ s = 9.29761+
Found by Joost de Winter
in September 2026. 		Σ s = 9.44501+
Found by Joost de Winter
in September 2026.

43.
           44.
           45.
Σ s = 9.54016+
Found by Joost de Winter
in September 2026. 		Σ s = 9.69537+
Found by Joost de Winter
in October 2026. 		Σ s = 9.81029+
Found by Joost de Winter
in October 2026.

46.
           47.
           48.
Σ s = 9.90032+
Found by Joost de Winter
in October 2026. 		Σ s = 9.98446+
Found by Joost de Winter
in September 2026. 		Σ s = 10.06189+
Found by Joost de Winter
in September 2026.

49.
           50.
Σ s = 10.14184+
Found by Joost de Winter
in September 2026. 		Σ s = 10.24031+
Found by Joost de Winter
in September 2026.