The final round of a contest had one knight (always tells the truth), one liar (always tells lies) and three knaves (alternate between truth and lies). The knight placed first and the liar placed fifth. Afterwards, each of the five contestants made two comments:

Alex: Bert wasn't first. Carl finished lower than Dave.

Bert: I was second. Alex finished lower than Eddy.

Carl: Alex finished higher off than Dave. Bert was fourth.

Dave: Eddy was second. Carl wasn't last.

Eddy: Bert wasn't first. I was third.

In what order did the contestants finish?

1st: Alex
2nd: Dave
3rd: Carl
4th: Eddy
5th: Bert

If the knight were to mention his own position, he would have to tell the truth and say that he finished first. Thus, neither Bert nor Eddy can be the knight.

Thus, Eddy's first statement is true (Bert is not the knight and so did not finish in first), so Eddy can't be a liar, either. Eddy is a knave, and he cannot have finished third.

Alex's first statement (the same as Eddy's) is true; he is either the knight or a knave.

Suppose Alex is the knight. Then, his second statement is true, and Carl finished lower than Dave.

Also, Alex must have finished first. That would make Bert's second statement false as well, and Bert has to be the liar, finishing last.

If Bert is in last place, Carl is not; Dave's second statement is true. We have already called Alex the liar, so Dave is a knave and his first statement is false. Thus Eddy is not second. We know he his not third (his own false statement), first (the knight is), or fifth (the liar is); Eddy finished in fourth place.

Alex (in first) did finish higher than Dave, and Bert was not in fourth; Carl is indeed the third Knave.

Carl and Dave are left in the second and third spots, but Alex the knight tells us that Carl finished lower than Dave.

So, the order that they finished in is:
1st: Alex (knight; T, T)
2nd: Dave (knave; F, T)
3rd: Carl (knave; T, F)
4th: Eddy (knave; T, F)
5th: Bert (liar; F, F)

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Just for completeness, let's say we assume Alex is a knave instead of the knight. Then, his first statement is still untrue, but his second is false; Dave finished lower than Carl.

Also, Eddy is a knave, and Bert cannot be the knight (he told at least one lie), so either Carl or Dave is the knight. Since Dave finished lower, the knight must be Carl.

Thus, Carl's statements must both be true. That puts Bert in fourth place and a knave, while Alex finished higher than Dave.

If Bert is a knave, and his first statement is false, his second must be true. That means that Alex finished lower than Eddy, but higher than Dave (from Carl's statements). Alex, Dave, and Eddy are already confined to the second, third, and fifth places; so Eddy is 2nd, Alex is third, and Dave is in fifth.

That would mean Dave should be the liar, but his first statement is true (Eddy is indeed in second). This cannot be, so the assumption that Alex is a knave was erroneous.

Therefore, the only valid solution is the previous one based on the assumption that Alex is the knight.

Posted by DJ on 2004-07-04 07:54:43