At a certain point the torturers claim that the cut off fingers could still be sewed back on by a good doctor. But that's more than 24 hours after the first finger, and thus impossible. The storing on ice is not very helpful either: the frost would damage the fingers after a short while.
The goof item below may give away important plot points.
In the end the torture victim and the torturer agree to write down a list of names simultaneously, in order to prove to each other that they both know these names, and thus are who they claim to be. Any other option to do so would be better. (i.e. you give me first names, I give you last names; you write down 1 name, I write down the 2nd, you the 3rd etc....) Because the opponents are both supposed to be very smart, neither one would agree to this.