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 = 6.89791+
Found by Joshua Wolk in September 2026.
s = 7
Trivial.
25.
26.
27.
s = 7.64575+
Found by Joshua Wolk in September 2026.
s = 7.79511+
Found by Joshua Wolk
in September 2026.
s = 7.95595+
Found by Yanis Gomes
in October 2026.
28.-32.
33.
34.
s = 8
Trivial.
s = 8.70711+
Found by Joshua Wolk
in September 2026.
s = 8.83512+
Found by Joshua Wolk
in September 2026.
35.
36.-40.
41.
s = 8.97730+
Found by Yanis Gomes
in October 2026.
s = 9
Trivial.
s = 9.62132+
Found by Joshua Wolk
in September 2026.
42.
43.
44.
s = 6+√14 = 9.74165+
Found by Grigory Merzon
in September 2026.
s = 9.89227+
Found by Grigory Merzon
in September 2026.
s = 9.96899+
Found by Yanis Gomes
in October 2026.
45.-50.
s = 10
Trivial.