The following pictures show n dominoes (with short side 1) packed inside the smallest known square (with side s).
1.-2.
3.-4.
5.
s = 2
Trivial.
s = 3
Trivial.
s = 2 + 5√2/4 = 3.76776+
Found by Jonathan Viquerat
in May 2026.
6.-8.
9.
10.-12.
s = 4
Trivial.
s = 4.86784+
Found by Aapo Lipponen
in July 2026.
s = 5
Trivial.
13.
14.
15.-18.
s = 5 + 1/√2 = 5.70710+
Found by Jonathan Viquerat
in May 2026.
s = 3 + 2√2 = 5.82842+
Found by Maurizio Morandi
in June 2026.
s = 6
Trivial.
19.
20.
21.-24.
s = 47/7 = 6.71428+
Found by Maurizio Morandi
in June 2026.
s = (9 + √23)/2 = 6.89791+
Found by Joshua Wolk
in September 2026.
s = 7
Trivial.
25.
26.
27.
s = 5 + √7 = 7.64575+
Found by Joshua Wolk
in September 2026.
s = 7.79482+
Found by Yanis Gomes
in October 2026.
s = 7.95092+
Found by Yanis Gomes
in October 2026.
28.-32.
33.
34.
s = 8
Trivial.
s = 8 + 1/√2 = 8.70710+
Found by Joshua Wolk
in September 2026.
s = (15 + √7)/2 = 8.82287+
Found by Grigory Merzon
in October 2026.
35.
36.-40.
41.
s = 8.95027+
Found by Grigory Merzon
in October 2026.
s = 9
Trivial.
s = 3(5+√2)/2 = 9.62132+
Found by Joshua Wolk
in September 2026.
42.
43.
44.
s = 9 + 1/√2 = 9.70710+
Found by Grigory Merzon
in October 2026.
s = 9.87098+
Found by Grigory Merzon
in October 2026.
s = 9.96212+
Found by Grigory Merzon
in October 2026.
45.-50.
s = 10
Trivial.