Squares in Triangles

The following pictures show n unit squares packed inside the smallest known equilateral triangles (of side length s).

1.
2.
3.

s = 1 + 2 / √3 = 2.154+
Trivial. s = 2 + 2 / √3 = 3.154+
Trivial. s = 3 / 2 + √3 = 3.232+
Found by Erich Friedman
in 1997.

4.
5.-6.
7.

s = 3 + 2 / √3 = 4.154+
Found by Erich Friedman
in 1997. s = 2 + 4 / √3 = 4.309+
Found by Erich Friedman
in 1997. s = 4 + 2 / √3 = 5.154+
Found by Erich Friedman
in 1997.

8.
9.
10.

s = 5.301+
Found by David W. Cantrell
in July 2002. s = 3 + 4 / √3 = 5.309+
Found by Erich Friedman
in 1997. s = 2 + 2 √3 = 5.464+
Found by Erich Friedman
in 1997.

11.
12.
13.

s = 5 + 2 / √3 = 6.154+
Found by Erich Friedman
in 1997. s = 6.301+
Found by David W. Cantrell
in July 2002. s = 4 + 4 / √3 = 6.309+
Found by Erich Friedman
in 1997.

14.-15.
16.
17.

s = 3 + 2 √3 = 6.464+
Found by Erich Friedman
in 1997. s = 6 + 2 / √3 = 7.154+
Found by Erich Friedman
in 1997. s = 7.301+
Found by David W. Cantrell
in July 2002.

18.
19.
20.-21.

s = 5 + 4 / √3 = 7.309+
Found by Erich Friedman
in 1997. s = 4 + 2 √3 = 7.464+
Found by Erich Friedman
in 1997. s = 3 + 8 / √3 = 7.618+
Found by Erich Friedman
in 1997.

22.
23.
24.

s = 8.15259+
Found by Markus Vesterinen
in May 2026. s = 8.301+
Found by David W. Cantrell
in July 2002. s = 6 + 4 / √3 = 8.309+
Found by Erich Friedman
in 1997.

25.
26.-27.
28.

s = 5 + 2 √3 = 8.464+
Found by Erich Friedman
in 1997. s = 4 + 8 / √3 = 8.618+
Found by Erich Friedman
in 1997. s = 3 + 10 / √3 = 8.773+
Found by Erich Friedman
in 1997.

29.
30.
31.

s = 8 + 2 / √3 = 9.154+
Found by Erich Friedman
in 1997. s = 9.301+
Found by David W. Cantrell
in July 2002. s = 7 + 4 / √3 = 9.309+
Found by Erich Friedman
in 1997.

32.
33.-34.
35.-36.

s = 6 + 2 √3 = 9.464+
Found by Erich Friedman
in 1997. s = 5 + 8 / √3 = 9.618+
Found by Erich Friedman
in 1997. s = 4 + 10 / √3 = 9.773+
Found by Erich Friedman
in 1997.

37.
38.
39.

s = 10.15299+
Found by Gianluca Costantino
in May 2026. s = 10.309+
Found by Emerson Connelly
in April 2026. s = 10.310+
Found by Emerson Connelly
in April 2026.

40.

s = 10.46426+
Found by Nathan Roll
in April 2026.

1.		2.		3.
s = 1 + 2 / √3 = 2.154+ Trivial.		s = 2 + 2 / √3 = 3.154+ Trivial.		s = 3 / 2 + √3 = 3.232+ Found by Erich Friedman in 1997.

4.		5.-6.		7.
s = 3 + 2 / √3 = 4.154+ Found by Erich Friedman in 1997.		s = 2 + 4 / √3 = 4.309+ Found by Erich Friedman in 1997.		s = 4 + 2 / √3 = 5.154+ Found by Erich Friedman in 1997.

8.		9.		10.
s = 5.301+ Found by David W. Cantrell in July 2002.		s = 3 + 4 / √3 = 5.309+ Found by Erich Friedman in 1997.		s = 2 + 2 √3 = 5.464+ Found by Erich Friedman in 1997.

11.		12.		13.
s = 5 + 2 / √3 = 6.154+ Found by Erich Friedman in 1997.		s = 6.301+ Found by David W. Cantrell in July 2002.		s = 4 + 4 / √3 = 6.309+ Found by Erich Friedman in 1997.

14.-15.		16.		17.
s = 3 + 2 √3 = 6.464+ Found by Erich Friedman in 1997.		s = 6 + 2 / √3 = 7.154+ Found by Erich Friedman in 1997.		s = 7.301+ Found by David W. Cantrell in July 2002.

18.		19.		20.-21.
s = 5 + 4 / √3 = 7.309+ Found by Erich Friedman in 1997.		s = 4 + 2 √3 = 7.464+ Found by Erich Friedman in 1997.		s = 3 + 8 / √3 = 7.618+ Found by Erich Friedman in 1997.

22.		23.		24.
s = 8.15259+ Found by Markus Vesterinen in May 2026.		s = 8.301+ Found by David W. Cantrell in July 2002.		s = 6 + 4 / √3 = 8.309+ Found by Erich Friedman in 1997.

25.		26.-27.		28.
s = 5 + 2 √3 = 8.464+ Found by Erich Friedman in 1997.		s = 4 + 8 / √3 = 8.618+ Found by Erich Friedman in 1997.		s = 3 + 10 / √3 = 8.773+ Found by Erich Friedman in 1997.

29.		30.		31.
s = 8 + 2 / √3 = 9.154+ Found by Erich Friedman in 1997.		s = 9.301+ Found by David W. Cantrell in July 2002.		s = 7 + 4 / √3 = 9.309+ Found by Erich Friedman in 1997.

32.		33.-34.		35.-36.
s = 6 + 2 √3 = 9.464+ Found by Erich Friedman in 1997.		s = 5 + 8 / √3 = 9.618+ Found by Erich Friedman in 1997.		s = 4 + 10 / √3 = 9.773+ Found by Erich Friedman in 1997.

37.		38.		39.
s = 10.15299+ Found by Gianluca Costantino in May 2026.		s = 10.309+ Found by Emerson Connelly in April 2026.		s = 10.310+ Found by Emerson Connelly in April 2026.