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 = 5.83555+
Found by Jonathan Viquerat
in October 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 = 6.53030+
Found by Pedro Martins
in October 2026.

22.
           23.
           24.
Σ s = 6.72926+
Found by Joost de Winter
in October 2026. 		Σ s = 6.86764+
Found by Jonathan Viquerat
in October 2026. 		Σ s = 7.04445+
Found by Jonathan Viquerat
in October 2026.

25.
           26.
           27.
Σ s = 7.18489+
Found by Joost de Winter
in October 2026. 		Σ s = 7.36392+
Found by Jonathan Viquerat
in October 2026. 		Σ s = 7.48522+
Found by Joost de Winter
in October 2026.

28.
           29.
           30.
Σ s = 7.65030+
Found by Joost de Winter
in October 2026. 		Σ s = 7.77414+
Found by Pedro Martins
in October 2026. 		Σ s = 7.93190+
Found by Pedro Martins
in October 2026.

31.
           32.
           33.
Σ s = 8.05629+
Found by Pedro Martins
in October 2026. 		Σ s = 8.21069+
Found by Pedro Martins
in October 2026. 		Σ s = 8.33193+
Found by Pedro Martins
in October 2026.

34.
           35.
           36.
Σ s = 8.47952+
Found by Pedro Martins
in October 2026. 		Σ s = 8.57769+
Found by Pedro Martins
in October 2026. 		Σ s = 8.71951+
Found by Pedro Martins
in October 2026.

37.
           38.
           39.
Σ s = 8.83526+
Found by Pedro Martins
in October 2026. 		Σ s = 8.97512+
Found by Joost de Winter
in October 2026. 		Σ s = 9.10166+
Found by Pedro Martins
in October 2026.

40.
           41.
           42.
Σ s = 9.22224+
Found by Joost de Winter
in October 2026. 		Σ s = 9.34186+
Found by Joost de Winter
in October 2026. 		Σ s = 9.46685+
Found by Joost de Winter
in October 2026.

43.
           44.
           45.
Σ s = 9.57992+
Found by Pedro Martins
in October 2026. 		Σ s = 9.70698+
Found by Pedro Martins
in October 2026. 		Σ s = 9.81537+
Found by Pedro Martins
in October 2026.

46.
           47.
           48.
Σ s = 9.94156+
Found by Pedro Martins
in October 2026. 		Σ s = 10.04215+
Found by Pedro Martins
in October 2026. 		Σ s = 10.15762+
Found by Pedro Martins
in October 2026.

49.
           50.
Σ s = 10.26929+
Found by Pedro Martins
in October 2026. 		Σ s = 10.37249+
Found by Pedro Martins
in October 2026.