Given a unit disk, find the smallest radius 
required for 
equal disks to completely cover the unit disk. The first few such values are
 |  |  | (1) |
 |  |  | (2) |
 |  |  | (3) |
 |  |  | (4) |
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
Here, values for 
, 8, 9, 10 are approximate values obtained using computer experimentation by Zahn (1962).
For a symmetrical arrangement with 
(known as the five disks problem), 
, where 
is the golden ratio. However, rather surprisingly, the radius can be slightly reduced in the general disk covering problem where symmetry is not required; this configuration is illustrated above (Friedman). Neville (1915) showed that the value 
is equal to 
, where 
and 
are solutions to
 | (11) |
 | (12) |
 | (13) |
 | (14) |
These solutions can be found exactly as
 |  |  | (15) |
 |  |  | (16) |
where
 | (17) |
 | (18) |
are the smallest positive roots of the given polynomials, with 
denoting the 
th root of the polynomial 
in the ordering of the Wolfram Language. This gives 
(OEIS A133077) exactly as
 | (19) |
where the root is the smallest positive one of the above polynomial.
is also given by 
, where 
is the largest real root of
 | (20) |
maximized over all 
, subject to the constraints
 | (21) |
 | (22) |
and with
 |  |  | (23) |
 |  |  | (24) |
 |  |  | (25) |
 |  |  | (26) |
 |  |  | (27) |
 |  |  | (28) |
 |  |  | (29) |
(Bezdek 1983, 1984).
Letting 
be the smallest number of disks of radius 
needed to cover a disk 
, the limit of the ratio of the area of 
to the area of the disks is given by
 | (30) |
(OEIS A086089; Kershner 1939, Verblunsky 1949).
