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 = 6.33218+
Found by Maksymilian Jankowski in April 2026. 		s = 43(5–√3)/22 = 6.387+
Found by Maurizio Morandi
in March 2026. 		s = 6.75549+
Found by Timo Berthold et al
in May 2026.

14.
           15.
           16.
s = 4√3 = 6.928+
Trivial.
  		s = 6.96367+
Found by Maksymilian Jankowski in April 2026. 		s = (19–5√3)/√2 = 7.311+
Found by Maurizio Morandi
in April 2026.

17.
           18.
           19.
s = 7.60383+
Found by Maksymilian Jankowski in April 2026. 		s = 32(9–2√3)/23 = 7.702+
Found by Maurizio Morandi
in April 2026. 		s = 25(3–√3)/4 = 7.924+
Found by Maurizio Morandi
in April 2026.

20.
           21.
           22.
s = 74(21–4√3)/131 = 7.948+
Found by Maurizio Morandi
in April 2026. 		s = 8.381+
Found by Jonathan Viquerat
in April 2026. 		s = 8.56767+
Found by Beppe Björkland
in October 2026.

23.
           24.
           25.
s = 5√3 = 8.660+
Found by Jonathan Viquerat
in April 2026. 		s = 8.795+
Found by Jonathan Viquerat
in April 2026. 		s = 30(4√3–3)/13 = 9.065+
Found by Jonathan Viquerat
in April 2026.

26.
           27.
           28.
s = 9.22075+
Found by Joshua Wolk
in September 2026. 		s = 9.27733+
Found by Bhavithran Ananthan
in July 2026. 		s = 45/2–15√3/2 = 9.50961+
Found by Yanis Gomes
in October 2026.

29.

s = 9.51666+
Found by Joshua Wolk
in September 2026. 		30.

s = 185√3/22–111/22 = 9.51951+
Found by Yanis Gomes
in October 2026. 		31.

s = 9.99495+
Found by Yanis Gomes
in October 2026.

32.

s = 10.17418+
Found by Joshua Wolk
in September 2026. 		33.

s = 10.33056+
Found by Joshua Wolk
in September 2026. 		34.

s = 10.39230+
Found by Joshua Wolk
in September 2026.

35.

s = 10.56385+
Found by Beppe Björkland
in October 2026. 		36.

s = 10.76836+
Found by Joshua Wolk
in September 2026. 		37.

s = 51/2–17√3/2 = 10.77756+
Found by Yanis Gomes
in October 2026.

38.

s = 10.83110+
Found by Joshua Wolk
in September 2026. 		39.

s = 10.84365+
Found by Joshua Wolk
in September 2026. 		40.

s = 48√3–72 = 11.13843+
Found by Yanis Gomes
in October 2026.

41.

s = 11.18027+
Found by Joshua Wolk
in September 2026. 		42.

s = 11.20046+
Found by Joshua Wolk
in September 2026. 		43.

s = 11.60446+
Found by Joshua Wolk
in September 2026.

44.

s = 11.75873+
Found by Joshua Wolk
in September 2026. 		45.

s = 11.90747+
Found by Beppe Björkland
in October 2026. 		46.

s = 12.03926+
Found by Joshua Wolk
in September 2026.

47.

s = 12.12435+
Found by Beppe Björkland
in October 2026. 		48.

s = 6039/314–1281√3/314 = 12.16637+
Found by Yanis Gomes
in October 2026. 		49.

s = 12.34679+
Found by Joshua Wolk
in September 2026.

50.

s = 12.36250+
Found by Joshua Wolk
in September 2026.