Rigid Circles in Squares
The following pictures show n unit circles packed inside the largest known square (of side length s) so that the circles are rigid.

1.
           2.
           3.
s = 2
Trivial. 		s = 2 + √2 = 3.414+
Trivial. 		s = 2 + 1/√2 + √6/2 = 3.931+
Trivial.

4.
           5.
           6.
s = 4
Trivial. 		s = 2 + √2 + √6 = 5.863+
Found by Erich Friedman
in 1998. 		s = 4 + √3 = 5.732+
Found by Erich Friedman
in 1998.

7.
           8.
           9.
s = 5.985+
Found by David Cantrell
in August 2006. 		s = 6.946+
Found by Craig Clapp
in August 2006. 		s = 7.15584+
Found by Yanis Gomes
in October 2026.

10.
           11.
           12.
s = 4+1/√2+√3+√6/2 = 7.66390+
Found by Yanis Gomes
in October 2026. 		s = 4 + √2 + √6 = 7.863+
Found by David Cantrell
in July 2002. 		s = 8.46991+
Found by Yanis Gomes
in October 2026.

13.
           14.
           15.
s = 8.992+
Found by Craig Clapp
in August 2006. 		s = 11/2+3√3/2+√(468+286√3)/26
s = 9.29185+
Found by Yanis Gomes
in October 2026. 		s = 9.50364+
Found by Yanis Gomes
in October 2026.

16.
           17.
           18.
s = 9.55592+
Found by Yanis Gomes
in October 2026. 		s = 10.55769+
Found by Yanis Gomes
in October 2026. 		s = 10.63978+
Found by Yanis Gomes
in October 2026.

19.
           20.
           21.
s = 10.88956+
Found by Yanis Gomes
in October 2026. 		s = 11.42181+
Found by Yanis Gomes
in October 2026. 		s = 11.28400+
Found by Yanis Gomes
in October 2026.

22.
           23.
           24.
s = 11.45568+
Found by Yanis Gomes
in October 2026. 		s = (226+67√3+9√(1243+544√3))/65
s = 11.73486+
Found by Yanis Gomes
in October 2026. 		s = 12.53689+
Found by Yanis Gomes
in October 2026.

25.
           26.
           27.
s = 13.26335+
Found by Yanis Gomes
in October 2026. 		s = 13.29859+
Found by Yanis Gomes
in October 2026. 		s = 2+3√2+2√6+√(1+2√2)
s = 13.09825+
Found by Yanis Gomes
in October 2026.

28.
           29.
           30.
s = 13.35269+
Found by Yanis Gomes
in October 2026. 		s = 13.77756+
Found by Yanis Gomes
in October 2026. 		s = 13.97250+
Found by Yanis Gomes
in October 2026.

31.
           32.
           33.
s = 2+4√2+2√6+√(1+2√2)
s = 14.51247+
Found by Yanis Gomes
in October 2026. 		s = 14.36761+
Found by Yanis Gomes
in October 2026. 		s = 2+4√2+2√6+√(1+2√2)
s = 14.51247+
Found by Yanis Gomes
in October 2026.

34.
           35.
           36.
s = 14.83164+
Found by Yanis Gomes
in October 2026. 		s = 14.71516+
Found by Yanis Gomes
in October 2026. 		s = 15.60219+
Found by Yanis Gomes
in October 2026.

37.
           38.
           39.
s = 15.59707+
Found by Yanis Gomes
in October 2026. 		s = 15.81558+
Found by Yanis Gomes
in October 2026. 		s = 15.25710+
Found by Yanis Gomes
in October 2026.

40.
           41.
           42.
s = 15.35967+
Found by Yanis Gomes
in October 2026. 		s = 15.65265+
Found by Yanis Gomes
in October 2026. 		s = 15.56828+
Found by Yanis Gomes
in October 2026.

43.
           44.
           45.
s = 15.77813+
Found by Yanis Gomes
in October 2026. 		s = 7+2√2+3√3+2√6/3
s = 16.65757+
Found by Yanis Gomes
in October 2026. 		s = 16.75332+
Found by Yanis Gomes
in October 2026.

46.
           47.
           48.
s = 16.94675+
Found by Yanis Gomes
in October 2026. 		s = 16.50781+
Found by Yanis Gomes
in October 2026. 		s = 16.85557+
Found by Yanis Gomes
in October 2026.

49.
           50.
s = 17.15392++
Found by Yanis Gomes
in October 2026. 		s = 17.15392+
Found by Yanis Gomes
in October 2026.