Triangles in Triangles

The following pictures show n equilateral triangles with side 1 packed inside the smallest known equilateral triangles (of side length s ). Most of these have not been proven.

1.		2.-4.		5.
s = 1 Trivial.		s = 2 Trivial.		s = 1 + √3 = 2.732+ Proved by Erich Friedman in 1997.

6.
7.-9.
10.

s = 13/8 + 3√13/8 = 2.977+
Found by Maurizio Morandi
in August 2008. s = 3
Trivial. s = 7/2 = 3.500
Found by Erich Friedman
in 1997.

11.
12.
13.

s = 9/4 + 9√21/28 = 3.722+
Found by Erich Friedman
in 1997. s = 2 + 2cos(π/9) = 3.879+
Found by David W. Cantrell
in July 2007. s = 3.992+
Found by Maurizio Morandi
in May 2008.

14.-16.
17.
18.

s = 4
Trivial. s = 4.465+
Found by Maurizio Morandi
in May 2008. s = 9/2 = 4.500
Found by Erich Friedman
in 1997.

19.
20.
21.

s = 14/3 = 4.666+
Found by Maurizio Morandi
in May 2008. s = 33/8 + 9√21/56 = 4.861+
Found by Maurizio Morandi
in May 2008. s = 4.923+
Found by Maurizio Morandi
in May 2008.

22.
23.-25.
26.

s = 4.996+
Found by Maurizio Morandi
in May 2008. s = 5
Trivial. s = 5.406+
Found by Joah Frenzley
in April 2026.

27.-28.
29.
30.

s = 11/2 = 5.500
Found by Erich Friedman
in 1997. s = 17/3 = 5.666+
Found by David W. Cantrell
in July 2007. s = 23/4 = 5.75
Found by Maurizio Morandi
in May 2008.

31.		32.		33.-36.
s = 5.912+ Found by Emerson Connelly in April 2026.		s = 5.951+ Found by Ian Watson in April 2026.		s = 6 Trivial.

37.		38.		39.-40.
s = 19/3 = 6.333+ Found by Ian Watson in April 2026.		s = 6.43454+ Found by Nathan Roll in April 2026.		s = 13/2 = 6.500 Found by Erich Friedman in April 2026.

41.		42.		43.
s = 20/3 = 6.666+ Found by Emerson Connelly in May 2026.		s = 27/4 = 6.750 Found by Emerson Connelly in May 2026.		s = 41/6 = 6.833+ Found by Emerson Connelly in May 2026.

44.		45.		46.-49.
s = 6.96238+ Found by Emerson Connelly in May 2026.		s = 6.98820+ Found by Emerson Connelly in May 2026.		s = 7 Trivial.

50.

s = 7.33527+
Found by Emerson Connelly
in May 2026.

6.		7.-9.		10.
s = 13/8 + 3√13/8 = 2.977+ Found by Maurizio Morandi in August 2008.		s = 3 Trivial.		s = 7/2 = 3.500 Found by Erich Friedman in 1997.

11.		12.		13.
s = 9/4 + 9√21/28 = 3.722+ Found by Erich Friedman in 1997.		s = 2 + 2cos(π/9) = 3.879+ Found by David W. Cantrell in July 2007.		s = 3.992+ Found by Maurizio Morandi in May 2008.

14.-16.		17.		18.
s = 4 Trivial.		s = 4.465+ Found by Maurizio Morandi in May 2008.		s = 9/2 = 4.500 Found by Erich Friedman in 1997.

19.		20.		21.
s = 14/3 = 4.666+ Found by Maurizio Morandi in May 2008.		s = 33/8 + 9√21/56 = 4.861+ Found by Maurizio Morandi in May 2008.		s = 4.923+ Found by Maurizio Morandi in May 2008.

22.		23.-25.		26.
s = 4.996+ Found by Maurizio Morandi in May 2008.		s = 5 Trivial.		s = 5.406+ Found by Joah Frenzley in April 2026.

27.-28.		29.		30.
s = 11/2 = 5.500 Found by Erich Friedman in 1997.		s = 17/3 = 5.666+ Found by David W. Cantrell in July 2007.		s = 23/4 = 5.75 Found by Maurizio Morandi in May 2008.