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.