Squares in Squares

The following pictures show n unit squares packed inside the smallest known square (of side length s ). For the n not pictured, the trivial packing (with no tilted squares) is the best known packing.

1.            2.            3.
s = 1
Trivial. 		s = 2
Proved by Frits Göbel
in 1979. 		s = 2
Proved by Frits Göbel
in 1979.

4.            5.            6.
s = 2
Trivial. 		s = 2 + 1 / √2 = 2.707+
Proved by Frits Göbel
in 1979. 		s = 3
Proved by Michael Kearney
and Peter Shiu in April 2002.

7.            8.            9.
s = 3
Proved by Erich Friedman
in 1999. 		s = 3
Proved by Erich Friedman
in 1999. 		s = 3
Trivial.

10.            11.            14.
s = 3 + 1 / √2 = 3.707+
Proved by Walter Stromquist
in 2003. 		s = 3.877+
Found by Walter Trump
in 1979. 		s = 4
Proved by Erich Friedman
in 1999.

15.            17.            18.
s = 4
Proved by Erich Friedman
in 1999. 		s = 4.675+
Found by John Bidwell
in 1997. 		s = (7 + √7) / 2 = 4.822+
Found by Pertti Hamalainen
in 1979.

19.            24.            26.
s = 3 + 4 √2 / 3 = 4.885+
Found by Robert Wainwright
in 1979. 		s = 5
Proved by Erich Friedman
in 1999. 		s = 7 / 2 + 3 / √2 = 5.621+
Found by Erich Friedman
in 1997.

27.            28.            29.
s = 5 + 1 / √2 = 5.707+
Found by Frits Göbel
in 1979. 		s = 3 + 2 √2 = 5.828+
Found by Frits Göbel
in 1979. 		s = 5.934+
Found by Thierry Gensane
and Philippe Ryckelynck
in April 2004.

37.            38.            39.
s = 6.598+
Found by David Cantrell
in September 2002. 		s = 6 + 1 / √2 = 6.707+
Found by Frits Göbel
in 1979. 		s = 6.818+
Found by David Cantrell
in August 2002.

40.            41.            50.
s = 4 + 2 √2 = 6.828+
Found by Frits Göbel
in 1979. 		s = 6.947+
Found by David Cantrell
in October 2005. 		s = 7.598+
Found by David Cantrell
in September 2002.

51.            52.            53.
s = 7 + 1 / √2 = 7.707+
Found by Frits Göbel
in 1979. 		s = 7 + 1 / √2 = 7.707+
Found by Frits Göbel
in 1979. 		s = 5 + 2 √2 = 7.823+
Found by David Cantrell
in September 2002.

54.            55.            65.
s = 7.848+
Found by David Cantrell
in October 2005. 		s = 7.987+
Found by David Cantrell
in October 2005. 		s = 5 + 5 / √2 = 8.535+
Found by Frits Göbel
in 1979.

66.            67.            68.
s = 3 + 4 √2 = 8.657+
Found by Evert Stenlund
in 1980. 		s = 8 + 1 / √2 = 8.707+
Found by Frits Göbel
in 1979. 		s = 15/2 + √7/2 = 8.822+
Found by David Cantrell
in September 2002.

69.            70.            71.
s = 8.856+
Found by David Cantrell
in September 2002. 		s = 8.912+
Found by David Cantrell
in August 2002. 		s = 8.963+
Found by David Cantrell
in October 2005.

82.            83.            84.
s = 6 + 5 / √2 = 9.535+
Found by Frits Göbel
in 1979. 		s = 4 + 4 √2 = 9.657+
Found by Evert Stenlund
in 1980. 		s = 9 + 1 / √2 = 9.707+
Found by Frits Göbel
in 1979.

85.            86.            87.
s = 11 / 2 + 3 √2 = 9.742+
Found by Erich Friedman
in 1997. 		s = 17 / 2 + √7 / 2 = 9.822+
Found by Erich Friedman
in 1997. 		s = 9.851+
Found by David Cantrell
in August 2002.

88.            89.
s = 9.901+
Found by David Cantrell
in August 2002. 		s = 5 + 7 / √2 = 9.950+
Found by Evert Stenlund
in 1980.

For more details, see my paper on the subject: Packing Unit Squares in Squares: A Survey and New Results.

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