alanback:
Yes, the assumption was that the generator has for each game a new initialisation. As you mentioned, there is also a second way possible, just one RNG for all games.
And only this second way is the right one - if Fencer implemented his RNG according to the first way, then
sooner or later somebody will count the cycle of the numbers...

Czuch:
There is a way, but a *very* hard one: The random numbers generators in programming languages are implemented as a function modulo m (for us with unknown m), in the simplest such random function there is a start number s, and the random numbers r1, r2, etc. are generated as s mod m =3 etc. There exist math proofs that random generators that include the modulo
function, even if they are not so simple as the one before, are always cyclical, i.e. after some thousand random numbers r1, r2, ..., rx they start from the beginning.
So if you play some very slow dice game like Anti Backgammon with a friend and the goal of both, not to win but to count the dice numbers, you may find out
a repating sequence of about 5000 - 6000 numbers. And
in a real game you can after 4 - 5 moves guess the next "random number". (Have not tested it, would be too hard effort just for winning some games...)