Q:

What are the clues in Pate de Foie Gras that should enable the reader to figure out how the Goose made Au-197?

Does Isaac Asimov give enough clues in Pate de Foie Gras for the reader to figure out the series of nuclear reactions that produced the gold eggs?
According to the link above:

In a commentary on the story, Asimov wrote that it was his intention
  for there to be a single solution discoverable by the reader. The hint
  dropped in the story is the description of an experiment in which the
  goose's gold production goes up when it is given water enriched with
  oxygen-18, which would indicate a possible source of the gold
  produced. This was expected to imply that if the goose is maintained
  in a closed environment, it will convert all the oxygen-18 to gold,
  while still being able to breathe the predominant oxygen nuclide
  (oxygen-16). It will excrete all the gold in its eggs, at which point
  it can be expected to start producing fertile eggs.

This is not a complete explanation, because the other strange thing about the goose was that it was not radioactive at all.  This means it had no Carbon-14 in its tissues.  As we know -- and knew in Asimov's time -- radioactive Carbon-14 is present in all living things.  (Also in dead tissue; see Radiocarbon Dating.)  
My question is not how could a molecular reactor (the Goose) sustain nuclear reactions.  Asimov never attempted to explain this; it is inexplicable. 
My question is:  did Asimov give other clues than those I have mentioned to explain how the disappearance of Carbon-14, plus a diet enriched in Oxygen-18 led to the production of Gold-197?  (I'm not asking a science question, I am asking about the content of a science fiction story.) 

A:

You're misinterpreting the statement about the puzzle.  The puzzle isn't how the hen creates gold.  The puzzle is how to allow the hen to produce fertile eggs.  The scientists hope that this would allow to breed other anomalous hens before the one amazing hen dies.  The narrator asks this puzzle in the story, and asks for readers in universe to submit solutions. 
The intended solution is probably to keep the hen in an environment lacking oxygen-18, but readers have found other solutions.  The hint for this solution is mentioned in the quote in your question.

