William asked in Science & MathematicsMathematics · 2 months ago

# Solve this hard Deduction Problem?

Jimmy claims that either Michelle or Toni tells the truth. Toni says that either Michelle or Jimmy tells the truth. Michelle says that neither Toni nor Jimmy tells the truth. Who, if anyone, is truthful?

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• atsuo
Lv 6
2 months ago

Let Jimmy = J, Michelle = M and Toni = T.

J says either M or T tells the truth.

T says either M or J tells the truth.

M says neither T nor J tells the truth.

Case1. J tells the truth

⇒ Exactly one of M or T tells the truth. So Case1-1 and Case1-2 exist.

Case1-1. M tells the truth

⇒ Both of T and J do not tell the truth.

But we know J tells the truth, it's a contradiction.

Case1-2. T tells the truth

⇒ Exactly one of M or J tells the truth.

We know J tells the truth, so M does not tell the truth.

That is, at least one of J or T tells the truth.

We know both of J and T tell the truth, no contradiction occurs.

Case2. T tells the truth

If we swap T and J then Case2 is similar with Case1.

Case3. M tells the truth

⇒ Both of T and J do not tell the truth.

T do not tell the truth, so both of M and J tell the truth or both of them do not tell the truth.

We know M tells the truth, so both of M and J tell the truth.

But we know both of J and T do not tell the truth, it's a contradiction.

Case4. No one tells the truth

M does not tell the truth, so at least one of J or T tells the truth.

But we know no one tells the truth, it's a contradiction.

Conclusion :

Jimmy and Toni are truthful, Michelle is not truthful.