A10. Amy, Bruce, Chris, Donna and Eve had a race. When asked in which order they finished, they all answered with a true and a false statement as follows:
Amy: Bruce came second and I finished in third place.
Bruce: I finished second and Eve was fourth.
Chris: I won and Donna came second.
Donna: I was third and Chris came last.
Eve: I came fourth and Amy won.
In which order did the participants finish?
Solution
Bruce, Donna, Amy, Eve, Chris

For convenience, let A1 denote Amy's first statement, B2 Bruces's second statement, and so on.

Suppose A1 is true so that Bruce was second. Then B1 is true and B2 is false. Then E1 is false and hence E2 is true, so that Amy won. However $\mathrm{C} 1$ and $\mathrm{C} 2$ are now both false, which is a contradiction.

Hence A1 is false so that A2 is true, i.e. Amy was third. Also, since B1 is now false, B2 is true so that Eve was fourth. Note that D1 is false, as Amy was third. So D2 is true and Chris came last. $\mathrm{C} 1$ is now false and so $\mathrm{C} 2$ is true, i.e. Donna came second. Finally, we deduce that Bruce won, giving the order Bruce, Donna, Amy, Eve, Chris.
(It is left to the reader to check that this order is consistent with each of the participants having given one true statement and one false statement.)