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.


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