The youngest cousin is 17 years old, and the oldest is 22 years old. Since there are six cousins, their ages must be 17, 18, 19, 20, 21, and 22 years. The cousin E, being the oldest, is 22 years old.

• A is older than B.
• A is one year older than C, so: $E > A > C > B$
• C is older than D, so: $E > A > C > B; C > D$

F’s age lies between B and D. This results in two possible age orders:

1. $E > A > C > B > F > D$
2. $E > A > C > D > F > B$

From these possibilities, F must be the second youngest, meaning F is 18 years old.