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.
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 = 7.61695+
Found by Haowei Lin
in July 2026.

21.
           22.
           23.
s = 3 + 8 / √3 = 7.618+
Found by Erich Friedman
in 1997. 		s = 7 + 2 / √3 = 8.15259+
Found by Markus Vesterinen
in May 2026. 		s = 8.301+
Found by David W. Cantrell
in July 2002.

24.
           25.
           26.-27.
s = 6 + 4 / √3 = 8.309+
Found by Erich Friedman
in 1997. 		s = 5 + 2 √3 = 8.464+
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.-34.
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.

35.-36.
           37.
           38.
s = 4 + 10 / √3 = 9.773+
Found by Erich Friedman
in 1997. 		s = 10.15299+
Found by Gianluca Costantino
in May 2026. 		s = 10.304+
Found by Gianluca Costantino
in April 2026.

39.
           40.
           41.-42.
s = 8 + 4 / √3 = 10.30940+
Found by Bhavithran Ananthan
in July 2026. 		s = 7 + 2√3 = 10.46410+
Found by Bhavithran Ananthan
in July 2026. 		s = 10.61880+
Found by Aristotle Papowitz
in July 2026.

43.
s = 10.77350+
Found by Aristotle Papowitz
in July 2026.