# Radius of circle coverage of n circles in square packing configuration

Is there a reference about determining the minimum radius of a circle that would cover n circles of radius 1 that are in a square packing configuration ( see Wolfram's MathWorld packing packing description)?

This is a different problem than the "best known packing of equal circles in a circle", though for n=1, 2, and 4 it would have the same result.

For a hexagonal packing configuration, n=1, 2, 3, 7, would have the same result as "best know packing of equal circles in a circle". Believe 6 also does, but would actually cover 7.

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## migrated from mathematica.stackexchange.comDec 22 '12 at 2:11

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For circles packing in square, it is solved problem. The formulas are here en.wikipedia.org/wiki/Circle_packing_in_a_square and here hydra.nat.uni-magdeburg.de/packing/csq/csq.html (there is a huge PDF there also to download) –  Nasser Dec 21 '12 at 19:38
Nasser: Not interested in circles fitting in a square. Interested in covering by a circle of n circles that are square packed. See diagram for what constitutes "square packing of circles". Klett: You are right it -- question probably should have gone there. Thought I was on that site until after I hit send. My apologies. –  Refactor Dec 21 '12 at 21:19

If you don't have a square number of small circles, you need to define which ones are missing. It is easy to say "the corners" if you have at least $n^2-4$, but what about if you have $89$ small circles? Asking for the minimum radius is fair.
A partial answer is that if you have a diagonal of $n$ unit circles, the length from one end to the other is $1+(n-1)\sqrt 2$, so the radius of the enclosing circle would be half that. The next layer would be if the farthest center is $n$ units one direction and $m$ units the other from the center (where $n,m$ can both be half integral or integral) the radius is $\frac 12(1+\sqrt{n^2+m^2})$ The minimum is just a constant increase over the size of the circle to contain a given number of lattice points. This is the Gauss circle problem, still unsolved.