Squares in Hexagons

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

1.
           2.
           3.
s = (3+√3)/6 = 0.778+
Trivial.
  		s = 2/√3 = 1.154+
Trivial.
  		s = (1+√3)/2 = 1.366+
Found by Maurizio Morandi
in March 2026.

4.
           5.
           6.
s = 2/31/4 = 1.519+
Found by Maurizio Morandi
in March 2026. 		s = √3 = 1.732+
Trivial.
  		s = 1.821+
Found by Maurizio Morandi
in March 2026.

7.
           8.
           9.
s = (√2)31/4 = 1.861+
Found by Maurizio Morandi
in March 2026. 		s = 2.15378+
Found by Ian Watson
in April 2026. 		s = 1/2+√3 = 2.23205+
Found by Jonathan Viquerat
in April 2026.

10.
           11.
           12.
s = 4/√3 = 2.309+
Trivial.
  		s = 2.40944+
Found by Jonathan Viquerat
in April 2026. 		s = 1+5√3/6 = 2.443+
Found by Maurizio Morandi
in March 2026.

13.
           14.
           15.
s = 2.572+
Found by Maurizio Morandi
in April 2026. 		s = 2.62126+
Found by Jonathan Viquerat
in April 2026. 		s = 2.75364+
Found by Grant Mowry
in May 2026.

16.
           17.
           18.
s = 2.82002+
Found by Spencer Leach
in May 2026. 		s = 2.87264+
Found by Jake Loyd
in May 2026. 		s = 2.92111+
Found by Ian Watson
in April 2026.

19.
           20.
           21.
s = 2.93511+
Found by Ian Watson
in April 2026. 		s = 3.16123+
Found by Ian Watson
in April 2026. 		s = (23+11√3)/13 = 3.23481+
Found by Spencer Leach
in May 2026.

22.
           23.
           24.
s = 3.31432+
Found by Jake Loyd
in June 2026. 		s = 3.36407+
Found by Jake Loyd
in June 2026. 		s = 3.40286+
Found by Thomas Greenleaf
in June 2026.

25.
           26.
           27.
s = 2√3 = 3.46410+
Trivial.
  		s = 3.49276+
Found by Jake Loyd
in June 2026. 		s = (9+7√3)/6 = 3.52072+
Found by Erich Friedman
in April 2026.

28.
           29.
           30.
s = 3.61979+
Found by Jake Loyd
in June 2026. 		s = 3.66437+
Found by Ian Watson
in April 2026. 		s = 3.69554+
Found by Ian Watson
in April 2026.

31.
           32.
           33.
s = 3.78143+
Found by Jake Loyd
in June 2026. 		s = 3.84140+
Found by Jake Loyd
in June 2026. 		s = 3.86634+
Found by Jake Loyd
in June 2026.

34.
           35.
           36.
s = 3.91853+
Found by Jake Loyd
in June 2026. 		s = 3.98201+
Found by Jake Loyd
in June 2026. 		s = 4.02042+
Found by Jake Loyd
in June 2026.

37.
           38.
           39.
s = 7/√3 = 4.04145+
Trivial.
  		s = 4.18036+
Found by Jake Loyd
in June 2026. 		s = 4.24512+
Found by Jake Loyd
in June 2026.

40.
s = 4.30173+
Found by Jake Loyd
in June 2026.