Most efficient algorithm to print 1-100 using a given random number generator We are given a random number generator RandNum50 which generates a random integer uniformly in the range 1–50.
We may use only this random number generator to generate and print all integers from 1 to 100 in a random order. Every number must come exactly once, and the probability of any number occurring at any place must be equal.

What is the most efficient algorithm for this?
Since other folks have given approximate solutions and solutions involving taking indeterminate numbers of deviates, how about a proof that there is no such algorithm that's guaranteed to only require a finite number of RandNum50() calls?

As others have noted, printing the numbers from 1-100 in random order is equivalent to printing a random permutation of these numbers; there are 100! of these permutations, and so any particular permutation must be output with probability $\frac{1}{100!}$.

But if we knew that our algorithm used at most $k$ calls to RandNum50 for some $k$, then we could argue as follows: firstly, pad out those computation paths that make fewer than $k$ calls to RandNum50 to make additional dummy calls (that is, calls where the returned value is irrelevant), so that all computation paths make precisely $k$ calls.  Any given sequence of $k$ results from our calls to RandNum50 must result in some output permutation, and so we can build an 'outcomes table' that maps any given sequence $(r_1, r_2, \ldots, r_k)$ of results from our calls into a particular output permutation.  Since each of these outcomes is equally likely (each of them has probability $\displaystyle\frac{1}{50^k}$), then the probability of getting any particular permutation out of our algorithm must be of the form $\displaystyle\frac{c}{50^k}$ for some $c$.  But $\displaystyle\frac{1}{100!}$ can't be of this form, because $100!$ doesn't divide $50^k$ for any $k$ (for instance, 3 divides $100!$ but can't divide any number of the form $50^k$).  This means that no possible distribution of outcomes to random-number calls can produce a uniform permutation.