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.

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

7.
8.
9.

s = 4 + 2 / √3 = 5.154+
Found by Erich Friedman
in 1997. s = 5.301+
Found by David W. Cantrell
in July 2002. s = 3 + 4 / √3 = 5.309+
Found by Erich Friedman
in 1997.

10.
11.
12.

s = 2 + 2 √3 = 5.464+
Found by Erich Friedman
in 1997. s = 5 + 2 / √3 = 6.154+
Found by Erich Friedman
in 1997. s = 6.301+
Found by David W. Cantrell
in July 2002.

13.
14.
15.

s = 4 + 4 / √3 = 6.309+
Found by Erich Friedman
in 1997. s = 3 + 2 √3 = 6.464+
Found by Erich Friedman
in 1997. s = 3 + 2 √3 = 6.464+
Found by Erich Friedman
in 1997.

16.
17.
18.

s = 6 + 2 / √3 = 7.154+
Found by Erich Friedman
in 1997. s = 7.301+
Found by David W. Cantrell
in July 2002. s = 5 + 4 / √3 = 7.309+
Found by Erich Friedman
in 1997.

19.
20.
21.

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

22.
23.
24.

s = 7 + 2 / √3 = 8.154+
Found by Erich Friedman
in 1997. 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.

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 = 4 + 8 / √3 = 8.618+
Found by Erich Friedman
in 1997.

28.
29.
30.

s = 3 + 10 / √3 = 8.773+
Found by Erich Friedman
in 1997. s = 8 + 2 / √3 = 9.154+
Found by Erich Friedman
in 1997. s = 9.301+
Found by David W. Cantrell
in July 2002.

31.
32.
33.

s = 7 + 4 / √3 = 9.309+
Found by Erich Friedman
in 1997. s = 6 + 2 √3 = 9.464+
Found by Erich Friedman
in 1997. s = 5 + 8 / √3 = 9.618+
Found by Erich Friedman
in 1997.

34.
35.
36.

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

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

7.		8.		9.
s = 4 + 2 / √3 = 5.154+ Found by Erich Friedman in 1997.		s = 5.301+ Found by David W. Cantrell in July 2002.		s = 3 + 4 / √3 = 5.309+ Found by Erich Friedman in 1997.

10.		11.		12.
s = 2 + 2 √3 = 5.464+ Found by Erich Friedman in 1997.		s = 5 + 2 / √3 = 6.154+ Found by Erich Friedman in 1997.		s = 6.301+ Found by David W. Cantrell in July 2002.

13.		14.		15.
s = 4 + 4 / √3 = 6.309+ Found by Erich Friedman in 1997.		s = 3 + 2 √3 = 6.464+ Found by Erich Friedman in 1997.		s = 3 + 2 √3 = 6.464+ Found by Erich Friedman in 1997.

16.		17.		18.
s = 6 + 2 / √3 = 7.154+ Found by Erich Friedman in 1997.		s = 7.301+ Found by David W. Cantrell in July 2002.		s = 5 + 4 / √3 = 7.309+ Found by Erich Friedman in 1997.

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

22.		23.		24.
s = 7 + 2 / √3 = 8.154+ Found by Erich Friedman in 1997.		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.
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 = 4 + 8 / √3 = 8.618+ Found by Erich Friedman in 1997.

28.		29.		30.
s = 3 + 10 / √3 = 8.773+ Found by Erich Friedman in 1997.		s = 8 + 2 / √3 = 9.154+ Found by Erich Friedman in 1997.		s = 9.301+ Found by David W. Cantrell in July 2002.

31.		32.		33.
s = 7 + 4 / √3 = 9.309+ Found by Erich Friedman in 1997.		s = 6 + 2 √3 = 9.464+ Found by Erich Friedman in 1997.		s = 5 + 8 / √3 = 9.618+ Found by Erich Friedman in 1997.

34.		35.		36.
s = 5 + 8 / √3 = 9.618+ Found by Erich Friedman in 1997.		s = 4 + 10 / √3 = 9.773+ Found by Erich Friedman in 1997.		s = 4 + 10 / √3 = 9.773+ Found by Erich Friedman in 1997.