Five friends met at a party and exchanged gossips. A said, “Only one of us is lying”. B said, “Exactly two of us are lying”. C said, “Exactly 3 of us are lying”. E said, “Exactly 4 of us are lying”. D said “All of us are lying”. Which one was telling the truth?
Question
Five friends met at a party and exchanged gossips. A said, “Only one of us is lying”. B said, “Exactly two of us are lying”. C said, “Exactly 3 of us are lying”. E said, “Exactly 4 of us are lying”. D said “All of us are lying”. Which one was telling the truth?
Solution
Let's analyze each statement:
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If A was telling the truth (only one is lying), then B, C, D, and E must be lying. But that would mean four people are lying, not one. So, A is lying.
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If B was telling the truth (exactly two are lying), then three people must be telling the truth. But if A, C, D, and E were telling the truth, then none of them would be lying, which contradicts B's statement. So, B is lying.
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If C was telling the truth (exactly three are lying), then two people must be telling the truth. If we assume C is one of them, then the other one could be A, B, D, or E. But A, B, and D's statements all contradict C's statement. E's statement also contradicts C's statement because if E was telling the truth, then four people would be lying, not three. So, C is lying.
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If D was telling the truth (all of us are lying), then D himself would be lying, which contradicts his own statement. So, D is lying.
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If E was telling the truth (exactly four of us are lying), then one person must be telling the truth. If we assume E is the one telling the truth, then A, B, C, and D must be lying. This is consistent with E's statement.
Therefore, E was the one telling the truth.
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