You own a machine that returns a meaningful number for "height Empire State Building" and "height average male". In less than a second. And which will perform floating-point division for you, at the press of a few keys. And you're vaguely unhappy because you can't do it all in one step!
Now I know how Hans Bethe (acknowledged grandmaster of the slide rule, who routinely computed logarithms in his head) must have felt when he heard the young grad students comparing the features on their scientific calculators.
I wonder if tools like Wolfram Alpha will eventually succeed in doing for Fermi problems what Google did for Trivial Pursuit: Turning an activity that was formerly regarded as a touchstone of intelligence into a typing exercise. Estimating the number of piano tuners in Chicago will be a lot less impressive when you can just type "piano tuners in Chicago" into a text box and get the answer back to within a decimal place.
Fortunately, right now typing "piano tuners in Chicago" into Alpha just makes Javascript crash in Safari, so I guess we humans can sleep soundly for a while longer. ;)
height Empire State Building divided by average height human
= 235
height Empire State Building divided by height human
Fails
a billion light years / average height human * (1/9) furlong / age universe = 1.01 c (speed of light)
Very neat.
Edit: And hitting "Show details" for a breakdown.