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Today, I went to grocery store (named H-E-B) and I got a irresistible offer. Buy 2 Nesquik cereal boxes and get a scholar kit (pencils, erasers, crayons, etc..) with the theme of the movie Kung Fu Panda 2 and also have the opportunity to get an iPad 2 if I win this little game:

Having $n$ players and a rectangular transparent box with measures $w$ for width, $l$ for length and $h$ for height find out how many balls (spouse balls are spherical) with radio $r$ are in the box if the box is full of balls. Each player have to give a guess, and wins the player who give the most correct answer. The correctness is giving in this way $|PlayerAnswer - RealAnswer|$, You win automatically if your correctness is $0$.

Then my question is: Does anyone have a good approach to solve this problem?

Special note: In the real game you don't know the measures.

Update: Video related

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So how many boxes of Nesquik cereal did you buy? – Leo Jun 25 '11 at 4:15
Do all players guess at the same time, or is there some order? Because therefore you could have a strategy depending on which guy you are in the ordering.. – Patrick Da Silva Jun 25 '11 at 4:23
@Leo, like I said I bought 2 boxes. @Patrick Da Silva, players guess in a different time. In fact is strange that 2 players can guess at the same time or even is some order. – razpeitia Jun 26 '11 at 2:23
up vote 7 down vote accepted

Wikipedia gives the approximate density of random packed spheres as $64\%,$ in contrast to the tightest packing of $\frac{\pi}{18}\approx 74\%.$ Good luck.

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According to ( my interpretation of Wikipedia's interpretation of) Song, C.; Wang, P. & Makse,H.A. (29 May 2008). "A phase diagram for jammed matter". Nature 453 (7195): 629–632, a random sphere packing can't exceed a density of 63.4 per cent ( So assume the density is, say, 63 per cent, and calculate the number of spheres on that assumption.

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