Problem #59: The King's Fortune Telling Scale
The king has a fortune telling scale. For one penny the scale spits out a slip
of paper with a fortune on it along with the weight of the king or whatever
is on the scale.
The king also has five bags of gold that he has given to five trusted friends
for safe keeping. Rumor has it that one of his five trusted caretakers is
not to be trusted and has been asking a counterfeiter to make phony gold.
All the king could find out about the counterfeit gold was that one of the
king's five bags is now filled with counterfeit gold and that the weight of
that bag is different from the others. He does not know whether it weighs
more or less than each of the others.
The king calls back all of his bags of gold. Using his fortune telling scale,
the king wants to find out:
Being a very economical king, he wants to find this out using the least number
of pennies possible. His wizard says it can be done with just 3 pennies.
No one else can see how. Anyone can do it with 5 pennies. Some say they can
do it with 4 pennies. Can you figure out how to do it with 3 pennies?
- which bag contains counterfeit gold, and
- how much it weighs, exactly.
(Assume that the fortunes the scale spits out offer no clues to which bag
contains the counterfeit gold.)