# The Labyrinthians' Exchange

by Kevin J. Lin

From Paul and Samantha's first two statements, we know the product, let's
call it P, isn't prime (because neither number was 1):

But much more informative is Paul's second statement that he can't figure
out where the party is with the information Samantha supplied. If P was
the multiple of exactly two prime numbers, Paul would know it by now (he'd
simply factor his number into its unique prime components). So he knows
the address isn't two prime numbers (although it could be one prime number
and a composite number). By stating this, he's telling Samantha as much.

Samantha's next statement is more telling still; she *already* knew
Paul couldn't figure it out. So whatever sum, let's call it S, Samantha
remembers, it can't be made by adding two prime numbers together- if it
could, then there would have been the possibility Paul knew the answer.

When Samantha announces her prescience, Paul then knows the answer. This
means that out of all the possible addresses that multiply to make P,
only one has the property of S described above.

Samantha makes the same logical deductions we have, and solves the problem.
once you've figured out the logical steps needed to solve the problem,
described above, it's still quite a bit of work to actually find the unique
pair of numbers that satisfies these properties. A computer
program can simplify the task of trial and error.

A more detailed discussion of the problem can be found in our Discussion
Forums.

Incidentally, the party's address is 4-13.