Given a unit disk, find the smallest radius 
required for 
equal disks to completely cover the unit
disk. The first few such values are
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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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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
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(11)
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(12)
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(13)
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(14)
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These solutions can be found exactly as
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(15)
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(16)
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where
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(17)
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(18)
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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
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(19)
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where the root is the smallest positive one of the above polynomial.
is also given by 
,
where 
is the largest real root of
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(20)
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maximized over all 
, subject to the constraints
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(21)
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(22)
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and with
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(23)
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(24)
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(25)
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(26)
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(27)
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(28)
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(29)
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(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
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(30)
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(OEIS A086089; Kershner 1939, Verblunsky 1949).
