Bilbo and Frodo are essentially counting 1 through 8 and then going
round again (with Tinker = 1, Tailor = 2, etc.). So if Bilbo
finished on Sailor = 4 and Frodo finished on Poor Man = 6, then
together they would reach $4+6=10$. But 10 is 2 after
8, and 2 = Tailor.
This problem is taken from the UKMT Mathematical Challenges.