Writing Project 31:

The old liar II

For all integers n and m we have the following problem P(n,m):

Three friends, Adam, Beth, and Cleo, are sitting together and talking. Together they are 150 years old.

Adam: n years ago, you two together were as old as I will be m years from now.
Beth: m years ago, you two together were as old as I will be n years from now.
Cleo: In m years, you two together will be as old as I am today.

If we know that the oldest, and only the oldest of the three is a liar, can you tell the ages of the three persons?

a) For which combination of n and m is there a solution to the problem P(n,m)?

b) For which combination of n and m are there more than one solutions?

c) Show that A is never the oldest of the three.

Erich Prisner, September 2003