The Contradictory Witness Statements
Four students - P, Q, R, and S - each made exactly one statement about who broke a window. Exactly one student is telling the truth; the other three are lying.
P says: "Q broke it." Q says: "S broke it." R says: "I did not break it." S says: "Q is lying."
No two students broke the window; exactly one culprit exists.
1. Determine who broke the window. Show how you test each candidate culprit against the 'exactly one truth-teller' rule.
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Test each candidate as culprit and count true statements.
If P broke it: P's claim (Q did) = false. Q's claim (S did) = false. R's claim (not R) = true. S's claim (Q lying) = true (Q is indeed lying). That gives 2 truths - invalid.
If Q broke it: P's claim = true. Q's claim (S did) = false. R's claim (not R) = true. S's claim (Q lying) = true. That gives 3 truths - invalid.
If R broke it: P's claim = false. Q's claim = false. R's claim (not R) = false. S's claim (Q lying) = true. That gives 1 truth - valid.
If S broke it: P's claim = false. Q's claim = true. R's claim = true. S's claim (Q lying) = false. That gives 2 truths - invalid.
Only R as culprit yields exactly one truth-teller. R broke the window, and S is the sole truth-teller.
2. Suppose the rule changes so that exactly two students tell the truth. Using the same statements, which culprit(s) would now be consistent?
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From the truth counts above: P → 2 truths; Q → 3 truths; R → 1 truth; S → 2 truths.
Under the 'exactly two truths' rule, both P and S are consistent culprits. The puzzle is under-determined: two different culprits satisfy the constraint, so the statements alone cannot identify a unique culprit. This illustrates that changing a constraint's value can destroy uniqueness even when the logic is unchanged.