[bigpooch]

Not exactly. The optimal strategy with no information about
the numbers that are seen gives the best probability of
being successful. If the writer is trying his best to make
it difficult for the picker, there is no question that the
aforementioned strategy will maximize the chances of picking
the correct number.

Look at some extreme cases when N = 1 billion. Suppose the
uniqueness restraint is lifted and after going through the
first 10 million slips of paper, you only find one digit
integers from 0 to 9 inclusive with seemingly uniform
distribution. It seems extremely likely that the largest
number will be 9!

Suppose N = 1 billion and the numbers are all unique but
after going through the first 10 million of them, it seems
that every number is an integer strictly between 1 and
1 billion and statistically resembles picking (without
replacement) an integer between 1 and 1 billion inclusive
(fortunately, the number 1 billion had not been picked yet).
It would seem very likely that the highest number will be
1 billion.

More generally, suppose N = 1 billion and after examining
the numbers of the first 10 million, it seems that these
real numbers come from a distribution function. If that
distribtution function is any well-known one, it seems
likely that the numbers do indeed come from that specific
distribution function so there would be a very good idea of
what the largest number could be.

Even if there were some noise introduced: for example, if
there were only a handful of ridiculously large or small
numbers among the other numbers which seem to come from
some probability distribution function, would you think that
you would have to sample 1000000000/e numbers?