Please look at Spooky math - with Perl. You will see that the numbers I have are parameters to the experiment, the guesser uses no knowledge about numbers, and the trick is that the guesser is sometimes guaranteed of being right.

The only limits to the guessing rule in that program are internal to how computers select pseudo-random numbers and the floating point math that Perl uses.

Other incidental notes. Your "trivial" uniform, infinite distribution actually does not and cannot exist. Its not existing has deep consequences. The details of why not are covered in real analysis. In the US and Canada this would traditionally be taken either by an advanced 4'th year math student or a beginning graduate student.

And random trivia. Not only is an infinite uniform distribution impossible, but attempts to look for really random numbers invariably turn up patterns that don't fit with a uniform distribution. For instance Benford's law states that the first digit obeys a logarithmic distribution. It isn't really a theorem, but other than that detail the following is a good introduction for the general public. Knuth tries to explain it in his series, but does not manage IMNSHO to show why his abstract model has anything to do with reality.

Just thought I would throw that out there...