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 = 2+10/3√3 = 3.92450+
Found by Johann Schellhorn
in July 2025. 		12.

s = 3.94164+
Found by Alpha Evolve
in May 2025. 		13.

s = 4
Trivial.
 

14.

s = 4.26899+
Found by Dominik Kamp
in April 2026. 		15.

s = 4.44727+
Found by Alex Kravatsky et al
in February 2026. 		16.

s = 4.52745+
Found by Aapo Lipponen
in October 2026.

17.

s = 4.6136+
Found by 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 = 5.39998+
Found by Jonathan Viquerat
in June 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 = 5.70971+
Found by Aapo Lipponen
in October 2026. 		27.

s = 5.71414+
Found by Aapo Lipponen
in October 2026.

28.

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

s = 5.99894+
Found by Joshua Wolk
in September 2026.
  		30-31.

s = 6
Trivial.
 

32.

s = 6.26029+
Found by Joshua Wolk
in September 2026. 		33.

s = 6.26579+
Found by Aapo Lipponen
in October 2026. 		34.

s = 6.34891+
Found by Joshua Wolk
in September 2026.

35.-37.

s = 6.35085+
Found by Joshua Wolk
in September 2026. 		38.

s = 6.76328+
Found by Joshua Wolk
in September 2026. 		39.

s = 6.92145+
Found by Joshua Wolk
in September 2026.

40.

s = 6.96222+
Found by Yanis Gomes
in October 2026. 		41.-43.

s = 7
Trivial. 		44.

s = 25√3/6 = 7.21687+
Found by Yanis Gomes
in October 2026.

45.

s = 7.37360+
Found by Joshua Wolk
in September 2026. 		46.

s = 7.41708+
Found by Joshua Wolk
in September 2026. 		47.

s = 7.46477+
Found by Joshua Wolk
in September 2026.

48.

s = 7.46748+
Found by Joshua Wolk
in September 2026. 		49.

s = 7.62458+
Found by Yanis Gomes
in October 2026. 		50.

s = 7.71830+
Found by Joshua Wolk
in September 2026.