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/

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