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A fun question I ask students or interviewees (in engineering) is:

This is not my question, this is an example:
Using only what you know now, how many cans of soda would you estimate are produced per day (on average) in the United States?

For this question, the result doesn't matter so much as the process you use.

In this theme of estimation, what's your favorite question?

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    http://en.wikipedia.org/wiki/Fermi_problem2010-07-30
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    [This](http://ocw.mit.edu/courses/mathematics/18-098-street-fighting-mathematics-january-iap-2008/) may be of interest.2010-07-30
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    I voted to close because I don't think this question is math-related. No math-related answers either.2010-07-31
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    In my defense...it was CW2010-08-03

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"How many estimation questions are asked in interviews across the world during a typical 24h period?"

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    I see what you did there...2010-07-30
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    I find it bothersome that out of all answers I posted on this site, this "smartass answer" is by far the most up-voted.2010-07-31
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One that I remember from some book (I think it was Innumeracy by John Allen Paulos) was "How fast does your hair grow, in miles per hour?"

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    This seems more like a factor-label problem to me, but I like it2011-04-08
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I like "how many licks does it take to get to the center of a tootsie roll pop".

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What is the mass of the atmosphere?