surprising fact about slicing up a pie

If you slice anything into N pieces of all different sizes what is the average size of the pieces?

1/N

avg =  (sum of piece sizes when expressed as a fraction of the total equals 1) / N

In other words:
avg =  (x/N + y/N + z/N + ... ) / (x+y+z+...)
avg = (x+y+z+...) / N / (x+y+z+...)  = 1/N

An infinity of pieces not included, but as Feynman pointed out, real objects can't be sliced up into an infinite number of pieces.

http://schneider.ncifcrf.gov/papers/avfrac/