The Labyrinthians' Exchange

by Kevin J. Lin

Two exceptionally quick-witted Labyrinthians were invited to a party in their honor. The party was to be held in a private house in the city of Mathematica. Mathematica, predictably, is laid out in a perfect grid. Every house's address is composed of two positive numbers, indicating its relative position to the Townhall in the southwest corner. For example, House 3-8 is two houses east of the Town Hall and seven houses north.

Currently Mathematica is one hundred by hundred houses in area.

Our two famed puzzlers, let's call them Samantha and Paul, were both given the address weeks in advance. And while both were quick-witted, they also were short memoried. Only a few hours before the party the following dialog occurred:

Paul: I'm afraid I've forgotten the address. I can only remember the product of the two numbers, and that the first number wasn't greater than the second.

Samantha: I've forgotten it too, but can only remember the sum of the two numbers, and that neither number was 1.

Paul: I can't figure out where the party is.

Samantha: I knew that.

Paul: OK, I know where the party is.

Samantha: OK, so do I.

Paul and Samantha were being perfectly truthful, and no other information was exchanged apart from this dialog.

What was the party's address?

See the Solution