Hexagons in Hexagons

The following pictures show n regular hexagons with side 1 packed inside the smallest known regular hexagon (of side length s ).

1.

s = 1
Trivial. 		2.

s = 2
Trivial. 		3.

s = 2
Trivial.

4.

s = 7/3 = 2.333+
Trivial. 		5.

s = 8/3 = 2.666+
Found by Erich Friedman
in March 2015. 		6-7.

s = 5/√3 = 2.886+
Found by Maurizio Morandi
in April 2015.

8.

s = (51+5√3)/18 = 3.314+
Found by Maurizio Morandi
in April 2015. 		9.

s = (24+5√3)/9 = 3.628+
Found by Maurizio Morandi
in April 2015. 		10.

s = 2 + √3 = 3.732+
Found by Maurizio Morandi
in April 2015.

11.

s = 3.92450+
Found by Timo Berthold et al
in January 2026.
Slightly improved by Alex Kravatsky et al
in February 2026.
 
  		12.

s = 3.94164+
Found by Alexander Novikov et al
in June 2025.
Improved slightly by Timo Berthold et al
in January 2026.
Slightly improved by Alex Kravatsky et al
in February 2026. 		13

s = 4
Trivial.
 
 
 
 
 

14.

s = 4.269+
Found by Timo Berthold et al
in January 2026.
Slightly improved by Alex Kravatsky et al
in February 2026. 		15.

s = 4.44727+
Found by Timo Berthold et al
in January 2026.
Slightly improved by Alex Kravatsky et al
in February 2026. 		16.

s = 4.52746+
Found by Timo Berthold et al
in January 2026.
Slightly improved by Alex Kravatsky et al
in February 2026.

17.

s = 4.6136+
Found by Timo Berthold et al
and Alex Kravatsky et al
in February 2026. 		18-19.

s = 8/√3 = 4.618+
Found by Maurizio Morandi
in April 2015.
  		20-21.

s = 5
Trivial.
 
 

22.

s = (39+8√3)/10 = 5.285+
Found by Maurizio Morandi
in April 2015. 		23.

s = 5.400+
Found by Alex Kravatsky et al
in February 2026. 		24.

s = 19√3/6 = 5.484+
Found by Maurizio Morandi
in April 2015.

25.

s = 5.62390+
Found by Alex Kravatsky et al
in February 2026. 		26.

s = 5.70972+
Found by Alex Kravatsky et al
in February 2026. 		27.

s = 5.71415+
Found by Alex Kravatsky et al
in February 2026.

28.

s = 5.90893+
Found by Alex Kravatsky et al
in February 2026. 		29-31.

s = 6
Trivial.