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There are many cases in maths where doing the same thing over and over again will bring you closer to a value. Derivation of polynomials is the striking example: no matter what you start with, you end up with 0. In a series expansion, the longer you continue the sequence the closer you get to the real value.

This concept applies in areas other than math (Wikipedia's everything-leads-to-philosophy phenomenon, for example). It's a little unrelated but it just proves the concept.

Hashing algorithms are just special math. Following logically, hash entropy supposedly decreases the more you hash a value. So is there a point at which any input value will give the same hash as any other?

(Let me just specify that we're not re-injecting any new data every time we hash, as if we're trying to increase the security of the hash by salting and re-hashing. It's strictly a matter of md5(md5(md5(... if that's the algorithm we're using.

I can see some problems with this already. Algorithms like md5 are designed to change their output radically when the input changes only by one bit of information. So even if a common hash for two values appears along the way, it itself will start changing. But then, does the fact that the output appears to be very different actually mean the possibilities are still many?

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You will end in circles. I believe the average circle length is $\sqrt{N}$ when there are $N$ possible values. There are many such circles, once again approximately $\sqrt{N}$. –  CodesInChaos Apr 23 '14 at 20:42
See Random Mapping Statistics; and this related answer. –  fgrieu Apr 23 '14 at 23:50

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