Question

114. Five persons, A, B, C, D and E, are either guards or thieves. The guards always tell the truth, whereas thieves always lie. A claims that B is a guard. B claims that C is a thief. C claims that D is a thief. E claims that A is a guard. D claims that B and E are different kinds. The number of thieves is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Start by assuming A is a guard, follow the chain of claims through B, C, D and E, and see if it loops back consistently; if it contradicts, flip A to a thief and try again.

Answer 1

The Correct Option is D

A neat way to crack this without guessing randomly is to notice the claims form a chain: A talks about B, B talks about C, C talks about D, D talks about B and E, and E talks about A. Following the chain from A in both of its possible states settles everyone at once.

1. If A is a guard: A's word is true, so B is a guard too. A truthful B then makes C a thief. A lying C then makes D's guilt claim false, so D is a guard. A truthful D then says B and E differ, and since B is a guard, E must be a thief. But a lying E means E's claim "A is a guard" is false, so A would have to be a thief, which contradicts where this branch started. This branch collapses.
2. So A is a thief. A's claim is false, so B is not a guard, meaning B is a thief. B's claim is then false, so C is not a thief, meaning C is a guard. C's claim is true, so D is a thief. D's claim is then false, so B and E are the same kind; since B is a thief, E is a thief too. Finally, E's claim is false, so A is not a guard, which agrees with A being a thief. No contradiction appears anywhere in this branch.

This second branch is the only one that survives, giving A = thief, B = thief, C = guard, D = thief, E = thief. That is four thieves and just one guard, C.

Let's summarize:

• Testing "A is a guard" leads to E's statement forcing A to be a thief, a direct contradiction.
• Testing "A is a thief" carries cleanly through all five claims without any contradiction.
• Only C ends up a guard; A, B, D and E are all thieves, so the count of thieves is 4.

So the number of thieves among the five people is 4.