Solution
Who Broke the Vase
Show the problem again
A vase was broken by exactly one of five children: Ana, Ben, Cal, Dee and Eli. Each child makes one statement, and exactly two of the five statements are true. - Ana: "It was not Ana." - Ben: "It was not Ben." - Cal: "Ana and Dee are not both telling the truth." - Dee: "Eli is lying." - Eli: "Cal is telling the truth." Who broke the vase? Answer with the name.
Worked solution
Let $T(X)$ mean that $X$ is truthful. Eli's statement says $T(\text{Eli}) = T(\text{Cal})$, and Dee's says $T(\text{Dee}) = \neg T(\text{Eli})$. So Cal and Eli have the same status and Dee has the opposite one. Suppose Cal and Eli are both truthful. Then they are the two truthful children, so Ana and Ben are lying, meaning the culprit is both Ana and Ben: impossible. Hence Cal and Eli lie and Dee is truthful. Cal's statement is false, so Ana and Dee are both truthful. That already makes two truthful children, so Ben's statement must be false: it was Ben. Checking all five culprits against all patterns with exactly two truthful children yields this as the only consistent scenario. Answer: Ben.
Source: Original puzzle in the classic 'who is lying' genre. Statement written for AxiomIQ.