Practice Questions: Binary Logic

Instructions 

Mohan went to an Island where there were three tribes of people. People from one of the tribes always spoke the truth. People from the second tribe always lied. People who belonged to the third tribe spoke truth and lie alternately. The Island had only 3 fruits : apple, mango and banana. Mohan met three people X,Y and Z asked them about their favourite fruits.
Their responses were as follows.
X: I like apple. Y likes mango
Y: Z likes mango. X likes banana
Z: I like apple. X likes banana
It is known that X, Y, and Z belong to different tribes and that each of them likes a different fruit. Answer the following questions.
Q1: Who likes apple? 
(a) X 
(b) Y 
(c) Z 
(d) cannot be determined
Ans: (b)
Explanation: Assume Y is the truth-teller. Then Y's statements - "Z likes mango" and "X likes banana" - must both be true. Therefore Y must like apple (since all three like different fruits). Check X's statements: "I like apple" is false and "Y likes mango" is false, so both of X's statements are false; this fits X being a knave (always lies). Z's two statements are "I like apple" (false) and "X likes banana" (true). These are one false and one true, which fits an alternator. All statements are thus consistent with Y being the truth-teller, X the knave and Z the alternator. Hence Y likes apple.

Q2: Who belongs to the tribe of truth tellers? 
(a) X 
(b) Y 
(c) Z 
(d) cannot be determined
Ans: (b)
Explanation: If Y is the truth-teller then Y's statements "Z likes mango" and "X likes banana" are both true. X's statements ("I like apple" and "Y likes mango") are both false, so X must be the knave. Z's statements ("I like apple" - false; "X likes banana" - true) are alternately false and true, so Z is the alternator. This assignment is consistent, so Y is the truth-teller.

Q3: Who belongs to the tribe of alternaters? 
(a) X 
(b) Y 
(c) Z 
(d) cannot be determined
Ans: (c)
Explanation: From the consistent assignment found above: Y is the truth-teller, X is the knave and Z is the alternator. Specifically, Z's first statement ("I like apple") is false and his second statement ("X likes banana") is true, showing an alternation of lie then truth. Thus Z is the alternator.

Q4: Ramesh, Suresh and Mahesh are three people who belong to three different tribes of people. The three tribes are known as knights (those who always speak the truth), Knaves(who always lie) and Alters(those who alternatively speak the truth and lie). Ramesh said that Suresh is not an alter. Mahesh said that Ramesh is an alter. 
Who among the following is Knave?
(a) Suresh 
(b) Ramesh 
(c) Mahesh 
(d) Cannot be determined
Ans: (a)
Explanation: Label the roles as (Ramesh, Suresh, Mahesh). Consider all six permutations of (Knight, Knave, Alter):
1. (T, L, A): Ramesh (T) says "Suresh is not an alter" - true (Suresh is L).Mahesh (A) makes only one statement. Since the starting point of an alternator (truth or lie) is unknown, this single statement may be either true or false. Therefore this case remains possible.
2. (T, A, L): Ramesh (T) would say "Suresh is not an alter" - false (Suresh is A), so impossible.
3. (L, T, A): Ramesh (L) would say "Suresh is not an alter" - as a lie this must be false, so Suresh would be an alter, but here Suresh is T, so impossible.
4. (A, T, L): Ramesh (A) could say "Suresh is not an alter" (true), but Mahesh (L) saying "Ramesh is an alter" would be a true statement (since Ramesh is A), which a knave cannot say, so impossible.
5. (L, A, T): Ramesh (L) saying "Suresh is not an alter" would be false (so Suresh is A), but Mahesh (T) saying "Ramesh is an alter" would then be false (Ramesh is L), so impossible.
6. (A, L, T): Ramesh (A) could say "Suresh is not an alter" (true, since Suresh is L), and Mahesh (T) saying "Ramesh is an alter" would be true. This permutation is possible.
Only permutations (T, L, A) and (A, L, T) remain possible, and in both cases Suresh is the knave. Hence Suresh is the Knave.

Q5: Ramesh, Suresh and Mahesh are three people who belong to three different tribes. The three tribes are known as knights (those who always speak the truth), Knaves(who always lie) and Alters(those who alternatively speak the truth and lie). Ramesh said that Suresh is not an alter. Mahesh said that Ramesh is an alter. 
Who is the Knight? 
(a) Suresh 
(b) Ramesh 
(c) Mahesh 
(d) cannot be determined
Ans: (d)
Explanation: From the analysis in Q4, two permutations remain possible: (Ramesh, Suresh, Mahesh) = (T, L, A) and (A, L, T). In the first, Ramesh is the Knight; in the second, Mahesh is the Knight. Both assignments satisfy the given statements, so it is not possible to determine uniquely which one is the Knight. Therefore the correct answer is "cannot be determined".