Hexagons in Squares

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+
Trivial.
s = 20(9–2√3)/23 = 4.813+
Found by Maurizio Morandi
in March 2026.

7.-8.
9.
10.

s = 3√3 = 5.196+
Found by Maurizio Morandi
in March 2026. s = (13–3√3)/√2 = 5.518+
Found by Maurizio Morandi
in March 2026. s = 35(4–√3)/13 = 6.106+
Found by Maurizio Morandi
in March 2026.

11.
12.

s = 5(3–√3) = 6.339+
Found by Maurizio Morandi
in March 2026. s = 43(5–√3)/22 = 6.387+
Found by Maurizio Morandi
in March 2026.

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+ Trivial.		s = 20(9–2√3)/23 = 4.813+ Found by Maurizio Morandi in March 2026.

7.-8.		9.		10.
s = 3√3 = 5.196+ Found by Maurizio Morandi in March 2026.		s = (13–3√3)/√2 = 5.518+ Found by Maurizio Morandi in March 2026.		s = 35(4–√3)/13 = 6.106+ Found by Maurizio Morandi in March 2026.

11.		12.
s = 5(3–√3) = 6.339+ Found by Maurizio Morandi in March 2026.		s = 43(5–√3)/22 = 6.387+ Found by Maurizio Morandi in March 2026.