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.


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