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?

Let -> be the logical notation for "implies".
Let E be the logical notation for "element of".
Let v be the logical notation for disjuction (OR).
Let ~ be the negation or complement operator (NOT).

Let "Cal restrained Abe" be denoted as X.
Let R denote the race, where R_final are the events occurring in the final part of the race, thus R = R_final v ~R_final.
Let A, B, and C denote Abe won, Ben won, and Cal won, respectively.

Abe¹: A -> ~(X E R_final)
Abe²: C

Ben¹: B
Ben²: X

Cal¹: B
Cal²: ~X

Given one and only one A, B, or C is True and the others are False.
Given that each are a different type and all three types, Knight, Knave, and Liar are represented, then

(1) If B is True and X is True, then Ben is a Knight, Cal is a Knave and Abe is a Liar.
(2) If B is True and X is False, then Ben is a Knave, Cal is a Knight and Abe is a Liar.
(3) If B is False and X is True, then Ben is a Knave, Cal is a Liar and Abe is a Knight.
(4) if B is False and X is False, then Ben is a Liar, Cal is a Knave and Abe is a Knight.

If Abe is a Liar, then the consequent of the 'material conditional' is False and the antecedant is True: ~A & ~(X E R_final)
If Abe is a Knight, then either the consequent of the 'material conditional' is True or OR both the antecedant and consequent are False.  Yet, given that only one person won and Abe's second statement were True, the consequent must be False, thus:
~A & (X E R_final).
In (3), X is True, and (X E R_final) is logically consistant in that it can be True or False, and being True in this case. In (4), X is False, thus (X E R_final) as True would not be logically consistant. Therefore there are three possible solutions:

(1) If Ben won the race is True and Cal restrained Abe is True, then Ben is a Knight, Cal is a Knave, Abe is a Liar, and Ben won the race.

(2) If Ben won the race is True and Cal restrained Abe is False, then Ben is a Knave, Cal is a Knight, Abe is a Liar, and Ben won the race.

(3) If Ben won the race is False and Cal restrained Abe is True, then Ben is a Knave, Cal is a Liar, Abe is a Knight, and Cal won the race.

Edited on April 13, 2013, 6:31 am

Posted by Dej Mar on 2012-02-01 06:03:18