Hexagons in Squares

The following pictures show n unit hexagons packed inside the smallest known square (with side s).

1.
           2.
           3.
s = (1+√3)/√2 = 1.931+
Trivial.
  		s = (1+2√3)/√2 = 3.156+
Trivial.
  		s = 9(6–√3)/11 = 3.491+
Found by Maurizio Morandi
in March 2026.

4.
           5.
           6.
s = (7–√3)/√2 = 3.725+
Found by Maurizio Morandi
in March 2026. 		s = 7(3–√3)/2 = 4.437+
Found by Maurizio Morandi
in March 2026. 		s = 20(9–2√3)/23 = 4.813+
Found by Maurizio Morandi
in March 2026.

7.-8.
           9.
           10.
s = 3√3 = 5.196+
Trivial.
  		s = (13–3√3)/√2 = 5.518+
Found by Maurizio Morandi
in March 2026. 		s = 6.10469+
Found by Jonathan Viquerat
in April 2026.

11.
           12.
           13.
s = 6.33235+
Found by Jonathan Viquerat
in April 2026. 		s = 43(5–√3)/22 = 6.387+
Found by Maurizio Morandi
in March 2026. 		s = 6.75558+
Found by Jonathan Viquerat
in April 2026.

14.
           15.
           16.
s = 4√3 = 6.928+
Trivial.
  		s = 6.96375+
Found by Jonathan Viquerat
in April 2026. 		s = (19–5√3)/√2 = 7.311+
Found by Maurizio Morandi
in April 2026.