Circles Covering Squares

The following pictures show n unit circles covering the largest known square (of side s ).

1. 2. 3.
s = √2 = 1.414+
Trivial. s = 4 / √5 = 1.788+
Trivial. s = 16 / √65 = 1.984+
Proved by Denes Nagy
in 1974.

4. 5. 6.
s = 2√2 = 2.828+
Proved by Denes Nagy
in 1974. s = 3.065+
Proved by Hans Melissen
in 1997. s = 3.347+
Found by Hans Melissen
in 1997.

7. 8. 9.
s = 1 + √7 = 3.645+
Proved by Hans Melissen
in 1997. s = 3.841+
Found by Hans Melissen
and P. Schuur in 1997. s = 4.335+
Found by Kari Nurmela
and Patric Östergård in 2000.

10. 11. 12.
s = (18 + 24√3) / 13 = 4.582+
Found by T. Tarnai
and Z Gáspár in 1995. s = 4.705+
Found by Hans Melissen
and P. Schuur in 1997. s = 4.943+
Found by Kari Nurmela
and Patric Östergård in 2000.

1.		2.		3.
s = √2 = 1.414+ Trivial.		s = 4 / √5 = 1.788+ Trivial.		s = 16 / √65 = 1.984+ Proved by Denes Nagy in 1974.

4.		5.		6.
s = 2√2 = 2.828+ Proved by Denes Nagy in 1974.		s = 3.065+ Proved by Hans Melissen in 1997.		s = 3.347+ Found by Hans Melissen in 1997.

7.		8.		9.
s = 1 + √7 = 3.645+ Proved by Hans Melissen in 1997.		s = 3.841+ Found by Hans Melissen and P. Schuur in 1997.		s = 4.335+ Found by Kari Nurmela and Patric Östergård in 2000.

10.		11.		12.
s = (18 + 24√3) / 13 = 4.582+ Found by T. Tarnai and Z Gáspár in 1995.		s = 4.705+ Found by Hans Melissen and P. Schuur in 1997.		s = 4.943+ Found by Kari Nurmela and Patric Östergård in 2000.