In a remote island, three people Abe, Ben and Cal took part in a running race. They belong to three different types: Knights, who always tell the truth; Liars who always lie and, Knaves who alternatively tell a truth or lie. It is known that precisely one of Abe, Ben and Cal is a knight, precisely one of them is a liar and the remaining is a knave. It is also known that there was precisely one person who reached the first position in the said race.

Following are some statements made by them:

Abe
1. I would have won the race if Cal had not come in my way to restrain me in the final part.
2. Cal won the race.

Ben
1. I won the race.
2. Cal came in Abe’s way, to restrain him from winning the race.

Cal
1. Ben won the race.
2. I did not come in Abe’s way, to restrain him from winning the race.

Find the types each of the three men belong to. Who won the race?

(In reply to re: three solutions? by scott)

With a little more reading up on material conditional statements, and noting that they can be TRUE even trivially. Abe's first statement is TRUE as the consequent "I (Abe) would have won the race" is to be considered TRUE.

If Ben is a Knight, due to Ben's first statement, Abe's second statement would be FALSE, thus Abe would be a Knave. This would mean Cal would be the Liar. Yet Cal's first statement is given as TRUE as it is logically the same as the first statement of Ben the Knight. Therefore, due to the contradiction, Ben could not be the Knight.

Similarly, if Cal is the Knight, due to Cal's first statement, Abe's second statement would be FALSE, and Abe would be a Knave. This would mean Ben would be the Liar. Yet Ben's first statement is given as TRUE as it is logically the same as the the first statement of Cal the Knight. Therefore, due to the contradiciton, Cal could not be the Knight.

If Abe were the Knight, then, by his second statement, Cal won the race. As Ben's and Cal's second statements contradict each other, one of them is the Knave and one is the Liar. Given that the consequent of Abe's first TRUE statement is TRUE, the antecedant may be either TRUE or FALSE and not change the truth-value of the statement.
Still, which of the two, Ben and Cal, is the Liar and which is the Knave can not be ascertained. Thus there are two possibilities:

(1) Abe is the Knight, Ben is the Knave, Cal is the Liar, and Cal won the race.
(2) Abe is the Knight, Ben is the Liar, Cal is the Knave, and Cal won the race.

It should be noted that "I won the race" and "I would have won the race" are not equivalent. The first is either TRUE or FALSE depending on whether (I) won the race or not, while the second is trivially TRUE.

Edited on April 13, 2013, 11:07 am

Posted by Dej Mar on 2013-04-13 10:46:45