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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.
           4.
s = 1
Trivial. 		s = 2
Proved by Frits Göbel
in 1979. 		s = 2
Proved by Frits Göbel
in 1979. 		s = 2
Trivial.

5.
           6.
           7.
           8.
s = 2 + 1 / √2 = 2.707+
Proved by Frits Göbel
in 1979. 		s = 3
Proved by Michael Kearney
and Peter Shiu in April 2002. 		s = 3
Proved by Erich Friedman
in 1999. 		s = 3
Proved by Erich Friedman
in 1999.

9.
           10.
           11.
           13.
s = 3
Trivial. 		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 Wolfram Bentz
in 2010.

14.
           15.
           17.
           18.
s = 4
Proved by Erich Friedman
in 1999. 		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 W. Cantrell
in September 2002. 		s = 6 + 1 / √2 = 6.707+
Found by Frits Göbel
in 1979. 		s = 6.818+
Found by David W. Cantrell
in August 2002.

40.
           41.
           50.
s = 4 + 2 √2 = 6.828+
Found by Frits Göbel
in 1979. 		s = 6.937+
Found by Joe DeVincentis
in April 2014. 		s = 7.598+
Found by David W. Cantrell
in September 2002.

51.
           52.
           53.
s = 7.704+
Found by Károly Hajba
in July 2009. 		s = 7 + 1 / √2 = 7.707+
Found by Frits Göbel
in 1979. 		s = 7.823+
Found by David W. Cantrell
in September 2002.

54.
           55.
           65.
s = 7.846+
Found by Joe DeVincentis
in April 2014. 		s = 7.954+
Found by Joe DeVincentis
in April 2014. 		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 W. Cantrell
in September 2002.

69.
           70.
           71.
s = 8.828+
Found by Maurizio Morandi
in June 2010. 		s = 8.881+
Found by Joe DeVincentis
in April 2014. 		s = 8.960+
Found by Joe DeVincentis
in April 2014.

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 W. Cantrell
in August 2002.

88.
           89.
s = 9.901+
Found by David W. 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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