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When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are randomly produced, and basically, the point of the game is to enter math equations to destroy targets in the least equations possible; writing an equation that intersects a glob means that it is destroyed. I haven't thought too much about this generalization but I thought it might be fun to examine, and I'm sure some work has been done in some field on a similar type question. The question I would eventually answer is the following :

"Let $R >0$ be a positive real number. Let $n >0$ be an integer. Fix some diameter $d and consider $n$ balls $\{B_i\}_{i=1}^n$ of diameter $d$ placed randomly inside the ball $B(0,R)$ of radius $R$ centered at the origin, such that each ball $B_i$ is contained entirely inside $B(0,R)$. (Intersections of the $B_i$ are acceptable). Fix some number $w$ denoting width.

Given $R,n,d,w$ as above, what is the average number of lines of width $w$ necessary so that at least one line intersects each ball in at least one point?"

Now, I'm sure that in it's full generality as above, the question is quite hard. Perhaps examining specific cases are easier or have already been done. Do you know of any literature on this subject or something similiar?

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    I loved this game in high school! I used to use graphs like $y=\tan(x)$ to get all of the globs in one shot. Sorry, no math content in this comment, just nostalgia.2013-03-18

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